ProbabilityDiscussion

October 7, 2010

Statistical Inference (Casella and Berger)

wikipedia:Set (mathematics)

wikipedia:Probability interpretations

http://en.wikipedia.org/wiki/Set_%28mathematics%29

http://en.wikipedia.org/wiki/Probability_interpretations

Set Theory:

Union - $A\bigcup B$ - combination of two sets

Intersection - $A\bigcap B$ - elements contained in both A and B

Complement - $A^{c}$ - everything that's not in A

Empty Set - Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \emptyset } - set containing no elements

Definitions:

experiment - any activity generating observable results

outcome - result of experiment (IMPORTANT TO KEEP STRAIGHT! don't confuse events and outcomes)

trial - single performance of experiment

sample space - set of all possible outcomes

countable/uncountable: - countable = one-to-one correspondence (e.g. 1/n) - uncountable = no one-to-one correspondence can be made -- infinite loop: you can do an infinite loop, but still count it -- flipping a coin: countable; temperature: uncountable

event - any subset of the sample space

Example:

experiment - roll a dice outcome - 1, or 2, or 3, or 4, or 5, or 6 trial - one roll of the dice COUNTABLE sample space - {1,2,3,4,5,6} event - may be {1}, or {1,2,3}, etc...

Operator Properties:

Commutative:

$A\bigcup B=B\bigcup A$

Associative:

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A\bigcup (B\bigcup C)=(A\bigcup B)\bigcup C}

Distributive:

$A\bigcap (B\bigcup C)=(A\bigcap B)\bigcup (A\bigcap C)$

DeMorgan's Law:

$(A\bigcup B)^{c}=A^{c}\bigcup B^{c}$

Call a set abnormal if it can be put into itself (otherwise it's normal)

Example: the set of all squares is not itself square, so it is not a member of the set of squares The complimentary set, containing all non-squares, is itself not a square, so is normal

Consider the set of all normal sets Is it normal or abnormal? If it were normal, it would be contained in itself, and would therefore be abnormal If it were abnormal, it would not be contained in itself, and would therefore be normal

You can resolve this using more rigorous set theory...

More Definitions:

Disjoint ("set" term) / mutually exclusive ("probability" term) - if the intersection of two sets is the null set, they are mutually exclusive

ie $A\bigcap B=\emptyset$

Partition - take a group of sets; if the union of these sets is the sample space, and they are mutually exclusive, this is a partition

ie $\bigcup _{i}^{\infty }A_{i}=S{\mbox{ and }}\bigcap _{i}^{\infty }A_{i}=\emptyset$

---

Distinction between probability theory that has a physical meaning (and is therefore "contaminated" by intuition) and a more abstract probability theory that doesn't have a corresponding physical meaning

Axiomatic probability theory (Komolgorov)

A probability is a function that follows 3 axioms:

Sample space $S$

(The domain) $\sigma$-algebra ${\mathfrak {B}}$ (means the set is fully consistent)

Function P -> probability over the domain ${\mathfrak {B}}$

1. $P(A)\geq 0$ for all $A\in {\mathfrak {B}}$

2. $P(S)=1$

3. If $A\in {\mathfrak {B}}$ and $B\in {\mathfrak {B}}$ are disjoint, then $P(A\bigcup B)=P(A)+P(B)$

In other words, Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P(\bigcup _{i=1}^{\infty }A_{i})=\sum _{l=1}^{\infty }P(A_{i})}

This is a mathematician's viewpoint: a clean definition, as long as we follow these rules, the function is a probability.

What is the probability of the null set?

Create a partition: $S={S,\emptyset }$

The probability of the sample space is $P(S)=1$

So $P(\emptyset )=1-P(S)=0$

$P(A)=1$

$P(A^{c})=1-P(A)$

If $A\subset B$ then $P(A)\leq P(B)$

The size of the set is directly related to the probability...

Another way to do this is using measure theory (another route, besides rigorous set theory, that leads to probability theory)

wikipedia:Measure theory

wikipedia:Sigma-algebra

Bonferroni's inequality: $P(A\bigcap B)\geq P(A)+P(B)-1$

Reading Assignment Discussion

Classical Definition of Probability (Laplace, 1812)

"If a random experiment can result in $N$ mutually exclusive and equally likely outcomes and if $N_{A}$ of these outcomes result in the occurrence of the event Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A} , the probability of Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A} is defined by $P(A)=P{A}={N_{A} \over N}$."

Example: rolling a dice

Event A might be how many times we roll a 1... or how many times we roll a 1 or a 2...

What if one side of the dice is weighted to favor 5? This definition doesn't work... No mathematical proof to show that the outcomes were mutually exclusive and equally likely.

Frequency

This is the limit, as the number of experimental trials performed (trials must be performed under "identical" conditions) goes to infinity, of the number of outcomes of the event of interest $N_{A}$:

$P(A)=lim_{n\rightarrow \infty }{N_{A} \over N}$

Limitations:

• can never actually perform an infinite number of trials
• what does "identical" mean? e.g. if you're performing a turbulence experiment, how can you initialize everthing the exact same way?
• (Tony): what's your infinity?
  • (Sean): whatever it is, it's not finite

Probability can be seen as lack of information (e.g. fluid mechanics is deterministic, so we could describe it if we know the state perfectly - we use probability to fill in/make up for the lack of knowledge)

Quantum theory: in two-split experiment, the very state of nature is random and needs to be defined using probability

Probability of an outcome in the future: judging confidence based on prior experience... statement of confidence

Bayesian vs. fequentist

Side discussion:

wikipedia:Noether's theorem

Advantages/Disadvantages of Axiomatic Approach

It provides a rigorous mathematical framework, removing bias/preference

But, when you get a result, you can't necessarily specify the meaning

October 11, 2010

Derivation of Bonferoni's Inequality

Show:

$P(A\bigcap B)\leq P(A)+P(B)-1$

Proof:

$P(A\bigcup B)=P(A)+P(B)-P(A\bigcap B)P(A\bigcap B)=P(A)+P(B)-P(A\bigcup B)$

But in general,

$P(A\bigcup B)\leq 1$

Plugging this back in,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(A \bigcap B) \leq P(A) + P(B) - 1 }

On Wikipedia: more general case; more than two sets

wikipedia:Boole's inequality

Conditional Probability and Bayes' Rule

if $A,B\in S$, and $P(B)\neq 0$,

$P(A|B)={\frac {P(A\bigcap B)}{P(B)}}$

And as a result, we get Bayes' Rule:

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P(A|B)=P(B|A){\frac {P(A)}{P(B)}}}

Derivation:

${\frac {P(A)}{P(B)}}\times P(B|A)={\frac {P(B\bigcap A)}{P(A)}}={\frac {P(A\bigcap B)}{P(A)}}\times {\frac {P(A)}{P(B)}}=P(A|B)$

Example application of conditional probability:

Run a combustion simulation... "Given that the temperature is in range X, what is the concentration range?"

Statistical Independence

Definition: Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P(A\bigcap B)=P(A)*P(B)}

Consequences:

$P(A|B)=P(A)$

Random Variable

wikipedia:Random variable

A random variable is a mapping from a sample space to real numbers.

Example: Dice roll

Mapping rolls to a set ${1,2,3,4,5,6}$

Example: Morse code

Looking at statistics of morse code...

Mapping dots and dashes to ${0,1}$ (or alternatively ${1,2}$)

IMPORTANT: Sample space is different from the random variable

Induced Probability Function

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(B)} for event B Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B \in \mathfrak{B}} on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S} , outcome Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_{i}}

(e.g. there's a sample space, and in the sample space there are events...

Before, we were talking about probability of events. Now, we're talking about probability of a random variable)

Random variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(s)} , random variable realization Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \chi} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A \subset \chi}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P_{X} (X \in A) = P{ S_{i} \in S : X(S_{i}) \in A }}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_{i}} are the outcomes that are in B

Cumulative distribution function

Definition:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_{X}(x) = P_{X} (X \leq x) \forall x}

Example: Dice roll

Probability of rolling a given number is constant (1/6)

Cumulative distribution function is a line, because for x=1, cumulative probability is 1/6; for x=2, cumulative probability is 2/6; and so on.

This definition is more general than the integral definition, because in general you need a cumulative distribution function that is differentiable

Sometimes there will be situations where a probability distribution function can't be defined, and only a cumulative distribution function can be defined

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_{X} \geq 0}

Further information:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \displaystyle{ \lim_{x \rightarrow -\infty} } = 0 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \displaystyle{ \lim_{x \rightarrow +\infty} } = 1 }

Example

Given 5% of men are colorblind, and 0.25% of women are colorblind: if a person is chosen at random and they are colorblind, what is the probability of their gender?

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} P(CB|M) &=& 0.05 \\ P(CB|W) &=& 0.0025 \\ P(M|CB) &=& ? \end{align} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(M|CB) = \frac{ P(M \bigcap CB)}{P(CB)} = \frac{ P(CB|M) P(M) }{ P(CB) } = \frac{ (0.05) (0.50) }{ P(M) P(CB|M) + P(CB|W)P(W) } = 0.952}

Identically Distributed

For two random variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X, Y} , and an event Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} : they are identically distributed if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X \in A) = P(Y \in A)}

We have an experiment, which we run a trial of, and we get an outcome. If the probability of that outcome is the same, the two outcomes are identically distributed.

The actual values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X, Y} are not important. e.g., probability of rolling a 2 on a dice is the same as rolling a 4 on a dice, so they are identically distributed, even thought 2 and 4 are not equal.

Probability Mass Function, Probability Density Function

Difference: one's continuous, one's discrete

Probability Mass Function (PMF): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X}(x) = P(X=x)}

• Discrete cases only

Probability Density Function (PDF): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X}(x) = \frac{ d }{ dx } F_{X}(x)}

• Continuous cases only

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_{X}} is the cumulative distribution function:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_{X}(x) = \displaystyle{ \int_{-\infty}^{x} f_{X}(x') dx' } }

Given properties of the CDF, what are the properties of the PDF?

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \displaystyle{ \int_{-\infty}^{\infty} f_{X}(x) dx } = 1 }

• If CDF is monotonically increasing, analogous property for PDF is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X}(x) \geq 0}

Above two properties are necessary and sufficient conditions for a function to be considered a PDF.

Further properties:

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(a \leq X \leq b) = \displaystyle{ \int_{a}^{b} f_{X}(x) dx } = F_{X}(b) - F_{X}(a) }

Example

Assume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda > 0} is a fixed positive constant

define function:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(x) = \begin{cases} 1/2 \lambda \exp(-\lambda x) & x \geq 0 \\ 1/2 \lambda \exp(\lambda x) & x < 0 \end{cases} }

Part a

What is the pdf of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(x)} ?

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(x) \geq 0 \qquad \forall x}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int_{-\infty}^{+\infty} f(x) dx &=& \int_{-\infty}^{0} f(x) dx + \int_{0}^{\infty} f(x) dx \\ & = & \frac{1}{2} \lambda \frac{1}{\lambda} \exp(\lambda x) \vert_{-\infty}^{0} + \frac{1}{2} \lambda ( - \frac{1}{\lambda} ) \exp(- \lambda x) \vert_{0}^{+\infty} \\ & = & 1 \end{align} }

Part b

Find probability that a random variable x is less than t, e.g. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(x<t) \quad \forall t \in \mathfrak{R}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(x < t) = \int_{-\infty}^{t} f(x) dx }

For this one, you have to integrate twice.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^{t} f(x) dx = \begin{cases} \frac{1}{2} \exp( \lambda t ) & t < 0 \\ 1 - {1}{2} \exp(- \lambda t) & t > 0 \end{cases} }

Another name for this is the cumulative distribution function.

Expectation

Definition:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E( g(x) ) = \displaystyle{ \int_{-\infty}^{+\infty} g(x) f_{X}(x) dx } }

Linearity:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E( a g_{1}(X) + b g_{2}(X) + c) = a E(g_{1}(X)) + b E(g_{2}(X)) + c }

Positivity:

If the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g(x) > 0 \forall x} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(g(X)) > 0}

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g_{1}(x) > g_{2}(x) \forall x} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(g_{1}(X)) > E(g_{2}(X))}

Moments:

nth moment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu_{n} = E( X^{n} )}

Central moments:

nth central moment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu^{\prime}_{n} = E( (X - \mu_{1})^{n} )}

2nd central moment: variance

Moment Generating Function (MGF)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M_{X}(t) = E( \exp( t X ) ) }

If you're looking at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M(-t)} , it's a Laplace transform. If it's Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M(-it)} , it's a Fourier transform.

wikipedia:Moment generating function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} M_{X}(t) &=& \int_{-\infty}^{+\infty} \exp( tx ) f_{X}(x) dx \\ &=& \int_{-\infty}^{+\infty} ( 1 + tx + \frac{t^2 x^2}{2!} + \frac{t^3 x^3}{3!} + ... ) f_{X}(x) dx \\ &=& \mu_{0} + t \mu_{1} + \frac{t^2}{2!} \mu_{2} + ... \\ \mu_{n} &=& M_{X}^{(n)}(0) \end{align} }

So knowing all the moments is equivalent to knowing the PDF.

October 14, 2010

Bernoulli Trial - two outcomes, known probability for each (e.g. heads or tails, or picking black and white marbles out of an urn)

Binomial distribution - probability of k successes in n Bernoulli trials

Normal distribution (and central limit theorem) - the sum of a bunch of random variables with same mean and variance approaches a Gaussian distribution

When doing an experiment - if there are a whole bunch of causes of error, all the errors add up, and you can expect a normal (Gaussian) distribution

October 18, 2010

Example 1: Expectation, CDF

Let x be a continuous non-negative random variable

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f} denote the PDF

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(x) = 0} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x < 0}

Show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(X) = \displaystyle{ \int_{0}^{\infty} \left( 1-F_{X} (x) \right) dx }}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x = \int_{0}^{x} 1 dx' }

Next, using definition of expectation... substitute this into the definition of the expectation.

(There's a problem - going from this step to the next step)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(X) = \int_{x' = 0}^{x' = \infty} \left[ \int_{x=x'}^{x=\infty} f(x) dx \right] dx' - \int_{x' = 0}^{x' = \infty} \left[ \int_{x=0}^{x=x'} f(x) dx \right] dx'}

where the first quantity in square brackets is 1, and the second quantity in square brackets is the CDF of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x'} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle = \int_{0}^{\infty} 1 dx' = \int_{0}^{\infty} F_{X} (x') dx }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle = \int_{0}^{\infty} ( 1 - F_{X} ) dx }

Example 2: Choosing Keys

A man has a set of N keys. He wants to open his door, which will open with exactly 1 key, but he doesn't know which one, and he is trying keys at random.

Part A

Find the mean number of trial attempts.

Use a negative binomial: http://en.wikipedia.org/wiki/Negative_binomial_distribution

Specifically, use a geometric distribution: http://en.wikipedia.org/wiki/Geometric_distribution

So we know that the expectation of the geometric distribution is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(X) = \frac{1}{p} }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p} is the probability of success in the Bernoulli trial,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p = \frac{1}{n}}

So that the mean number of trial attempts is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(X) = n}

Part B

What if there is no replacement?

With no attempts,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X = 0)=0}

After the first attempt,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X=1) = \frac{1}{n}}

After the second attempt,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X=2) = \frac{ (n-1) }{ n (n-1) }}

And the third attempt,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X=3) = \frac{ (n-1)(n-2) }{ n (n-1) (n-2) }}

And so on. Each time, all terms cancel out except

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X=x) = \frac{1}{n}}

So we can take the expectation of that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(X) = \displaystyle{ \sum_{x=1}^{n} } \frac{x}{n}}

And using the formula for the first Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} counting numbers,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(X) = \frac{1}{n} \frac{n (n+1)}{2}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(X) = \frac{n+1}{2}}

Multivariate

Set theory

Moved into random variables (mapping from event space to the real numbers)

Now we want to do a new type of mapping into multiple random variables

n-dimensional random vector - a mapping from a sample space into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{R}^n} Euclidian space.

Joint Probability Mass Function: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X,Y,\dots} (x,y,\dots) = P(X=x, Y=y, \dots )}

Joint Probability Density Function: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P( {X, Y, \dots \in A} ) = \int_{A} \dots \int f_{X,Y,\dots} (x,y,\dots) dx dy \dots}

Joint Cumulative Distribution Function: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X \leq x, Y \leq y, \dots ) = F_{X,Y,\dots} (x,y,\dots) = \int_{0}^{x} \dots \int_{0}^{y} f_{X,Y,\dots} (x,y,\dots) dx dy \dots }

Question: Why is joint PDF defined in terms of P, whereas the univariate PDF is defined in terms of the CDF?

Answer: Boundaries of multivariate PDFs are often non-trivial, and are not nice even "rectangles"... You need to know the boundaries of the PDF really well to use the CDF, so the joint CDF is not used as often.

Joint Conditional PDF: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X|Y} (x|y) = P(X=x|Y=y) = \frac{ f_{X,Y} (x,y) }{ f_{Y} (y) }}

The conditional PDF is just a renormalization.

But how is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{Y} (y)} defined, if it's a multivariate PDF?

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{Y} (y) = \int_{x=-\infty}^{\infty} f_{X,Y} (x,y) dx }

This is the marginal PDF...

Marginal PDF: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X} (x) = \int_{Y} \dots \int_{Z} f_{X,Y,\dots,Z} (x,y,\dots,z) dz \dots dy }

Definition of independence of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Y} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X,Y} (x,y) = f_{X} (x) f_{Y} (y) }

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P ( { X \in A, Y \in B } ) = P({ X \in A}) * P({ Y \in B }) }

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(g(X) h(Y)) = E(g(X)) * E(h(Y))}

• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z = X + Y} , then the moment generating function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M_{Z} (t) = M_{X} (t) * M_{Y} (t)}
  • This was used in deriving Central Limit Theorem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z = X+Y ~ Norm( \mu{x} + \mu{y}, \sigma_{x}^2 + \sigma_{y}^2 )}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X ~ Norm(\mu_{x}, \sigma_{x}^2)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Y ~ Norm(\mu_{y}, \sigma_{y}^2)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(X) = E(E(X|Y))}

Define a new variable: covariance

Covariance: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Cov(X,Y) = E((X-\mu_{x})(Y-\mu_{y})) = E(XY) - \mu_{X} \mu_{Y} }

Correlation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho_{XY} = \frac{ Cov(X,Y) }{\sigma_{X} \sigma_{Y}}}

Note: Just because the covariance is 0 does not mean that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Y} are independent, i.e. it doesn't imply Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(XY) = E(X) E(Y)}

Bivariate Normal Distribution

• Means Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu_{X}, \mu_{Y}}
• Variances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sigma_{X}^2, \sigma_{Y}^2}
• Correlation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X,Y} (x,y) = \displaystyle{ \frac{ \exp{ \frac{-1}{2(1-\rho^2)} \left[ \left( \frac{x-\mu_{x}}{\sigma_{x}} \right) - 2 \rho \left( \frac{x - \mu_{x}}{\sigma_{x}} \right) \left( \frac{y-\mu_{y}}{\sigma_{y}} \right) + \left( \frac{y-\mu_y}{\sigma_y} \right)^2 \right] } }{ 2 \pi \sigma_x \sigma_y \sqrt{ 1 - \rho^2 } } } }

Transform of a PDF

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{X}, f_{ \overrightarrow{X} } ( \overrightarrow{x} )}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{ U } \rightarrow U_{1} = g_{1} ( \overrightarrow{X} ), U_{2} = g_{2} ( \overrightarrow{X} ), \dots }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h = g^{-1}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J = det \left( \left[ \frac{\partial h_i}{\partial u_j} \right]_{i,j} \right)}

The transformed PDF is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{\overrightarrow{U}} (\overrightarrow{u}) = f_{x} \left[ h_1 (\overrightarrow{u}), h_2 (\overrightarrow{u}), \dots \right] \cdot \vert J \vert}

Multivariate Example 1

X and Y have the distribution:

		X
		1	2	3
Y	2	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{12}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{6}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{12}}
	3	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{6}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{6}}
	4	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{3}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0}

Part A

Show that X and Y are not independent.

One way: show that the covariance is nonzero.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(XY) = \displaystyle{ \sum_{i=1}^{3} \sum_{j=2}^{4} x_{i} y_{j} f_{X,Y} } }

A more fundamental way: using the definition of independence

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X,Y} (x,y) =? f_{X} (x) * f_{Y} (y) }

So sum up each row/column and put it in a new row/column

		X
		1	2	3	(sum)
Y	2	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{12}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{6}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{12}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{3}}
	3	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{6}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{6}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{3}}
	4	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{3}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{3}}
	(sum)	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{4}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{4}}

Then show that the product of the two marginal PDFs is not equal to the joint PDF value

i.e. pick row i and column j, and if the sum of the joint PDF across the whole row i, times the sum of the joint PDF across the whole column j, does not equal the joint PDF at location (row i col j), then we know the definition of independence is not met

Part B

Give a probability table for random variables U and V with the same marginals as X and Y but are independent.

So, we want to keep the "sum" column and row. Then we want to multiply the sum for row i by the sum for column j,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_{U,V} = f_{U} * f_{V} }

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f_{U}=f_{X};f_{V}=f_{Y}}

U is a discrete binomial distribution

V is uniformly distributed

Notes

If they're independent, they WILL have a zero covariance

(So it follows that, if the covariance is nonzero, there is no way they can be independent)

But, just because the covariance is zero doesn't mean they are independent

October 21, 2010

Review

Hypergeometric - out of n objects, picking k objects (bernoulli trials) without replacement, the probability that x of them are success

Binomial distribution - out of n objects, picking k objects (bernoulli trials) with replacement, the probability that x of them are success

Normal distribution - derived using Central Limit Theorem; central concept is, if you take an infinite number of random variables distributed with the same mean and variance, the sum of these infinite number becomes a normal distribution

Independence of random variables - the joint PDF is equal to the products of the marginal PDFs

Joint normal distribution - determinant of covariance matrix shows up

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Sigma = \left[ \begin{array}{cc} \sigma_{X_1} & \sigma_{X_1 X_2} \\ \sigma_{X_1 X_2} & \sigma_{X_2} \end{array} \right] }

If we then find the eigenvalues of this matrix, and use these as the covariance matrix (diagonal), then some ellipsoidal, skewed distribution would become a normal distribution composed of nice circles.

Also: can define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Y = \Sigma^{-1} [X - \mu]} to center the distribution around the means.

What if you have a kidney-shaped distribution?

You can define a new variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Y} , and use the covariance matrix, and get a "rotated" distribution, centered around the means, that's sort-of circular. But the kidney shape will still remain.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sigma_{Y_j Y_i} = 0, i \neq j}

Covariance (correlation) is 0, but the two variables are not independent. (i.e. using mathematical trick to make the variance 0, but the joint distribution is not equal to the product of the two marginal PDFs).

You cannot make the distribution fit into a circle, because then you're throwing away information that's important.

To first order, you may be able to approximate it as a joint-normal distribution. (But this is like approximating a human as a sphere).

If you have an ellipsoidal distribution that's extremely squished in one direction,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_2 \approx g(X_1) \approx a X_1}

Homework Problem 1

Let the number of chocolate chips in a certain type of cookie have a Poisson distribution. We want the probability that a randomly chosen cookie has at least two chocolate chips to be greater than 0.99. Find the smallest value of the mean of the distribution that ensures this probability.

Poisson distribution:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X=x) = \frac{ \lambda^{x} e^{-\lambda} }{ x! } }

The question is asking for: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P({X \geq 2}) > 0.99}

Rewriting using a less-than sign: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P({ X < 2 }) < 0.01}

What is the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda} that satisfies

CDF: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(x) = \frac{ \Gamma( x+1 , \lambda ) }{ x! }} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda} is the mean and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Gamma} is the incomplete Gamma function

We want to plug 2 into the CDF, set it equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0.01} , and do a nonlinear solve to find values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda}

We know Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x < 2} , so it can be 0 or 1, and plugging this into expression for Poisson distribution:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ \exp{(-\lambda)} \lambda^{0} }{ 0! } + \frac{ \exp{(-\lambda)} \lambda^{1} }{ 1! } < 0.01}

(replace the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle <} sign with an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =} sign... and solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda} ...)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda = 6.638 }

Homework 2

Two movie theaters compete for the business of one thousand customers. Assume that each customer chooses between the movie theaters independently and with indifference (p=1/2). Find and expression for N, the number of seats in your theater, that will result in the probability of turning away a customer (due to filling the seats) being less than one percent. First use the discrete distribution and then use the continuous approximation.

Problem is asking for N, such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P({K > N}) < 0.01 }

Binomial distribution: "What is the probability that n identical but independent bernoulli trials result in k successes?"

• Are we doing this with replacement or not?
• P stays the same - probability is 1/2 for each binomial trials (whether someone picks our movie theater or not)
• So that's like having replacement

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle K \sim Binomial(n=1000, p=0.5) = \left( \begin{array}{c} n \\ k \end{array} \right) p^k (1-p)^{n-k} }

Want the probability of everything under the curve, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle K > N} :

For the CDF: since it is discrete, the actual value will fall somewhere between two discrete values. Which one do we use?

• we want to pick the one to the right - to make sure we have an extra seat

Using binomial distribution web app, n=1000, p=0.5: plots PDF and CDF)

Number of seats: 537 (makes sense - half of 1000 is 500, and if it's a random night,

Part 2:

How do we do this with the normal (continuous) distribution? How would d'Alembert do this?

Homework 3

For the joint pdf Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X,Y}(x,y) = c(x+2y)} over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0<y<1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0<x<2} (the pdf is zero elsewhere), find the coefficient c.

Also, find the joint CDF and the two marginal PDFs.

First part: The integral over the entire domain must be 1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c = \frac{1}{4}}

Second part:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_{X,Y} (x,y) = }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{1}{4} xy (x+y) & \dots & 0 < x < 2, 0 < y < 1 \\ \frac{1}{4} x(\frac{1}{2}x + 1) & \dots & 0 < x < 2, y \geq 1 \\ \frac{1}{2} y (1+y) & \dots & x \geq 2, 0 < y < 1 \\ 1 & \dots & x \geq 2, y > 1 \\ 0 & \dots & elsewhere \end{align} }

Challenge: define a new variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = \frac{9}{ (X+1)^2 }} , and find the marginal PDF of z.

Homework 4

For two independent random variables X and Y with moment generating functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M_X(t)} and M_Y(t), show that the sum of these variables, Z = X+ Y, has a moment generating function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M_Z(t) = M_X(t) M_Y(t)} .

Suggestion: if you first show that for independent variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(g(X) h(Y)) = E(g(X)) E(h(Y))} then you can use the "cool" way of writing the moment generating function to give this proof in about four steps.

Given: definition of independence

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X,Y} (x,y) = f_{X}(x) f_{Y}(y) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(g(X) h(Y)) = \int_{-\infty}^\infty \int_{-\infty}^\infty f_{X,Y}(x,y) g(X) h(Y) dx }

since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X,Y} (x,y) = f_{X}(x) f_{Y}(y) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(g(X) h(Y)) = \int_{-\infty}^\infty \int_{-\infty}^\infty f_{X} f_{Y} g(X) h(Y) = \int_{-\infty}^\infty f_X g(X) dx \int_{-\infty}^\infty f_Y h(Y) dy = E(g(X)) E(h(Y)) }

Now, let's look at two particular functions for g and h:

Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g(X) = \exp{(tX)} } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h(Y) = \exp{(tY)} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E( \exp{(tX))} E( \exp{ (tY) } ) = E( \exp{(tX)} \exp{(tY)} ) = E( \exp{(t (X+Y)) } ) = E( \exp{(tZ)} ) = M_Z(t) }

October 25, 2010

Conditional Expectation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle g(X) \vert Y \rangle = \displaystyle{ \int_{-\infty}^{\infty} g(x) f_{X,Y} dx = \int_{-\infty}^{\infty} g(x) \frac{ f_{X,Y} }{ f_{Y} } dx = \frac{ 1 }{ f_{Y} } \int_{-\infty}^{\infty} g(x) f_{X,Y} dx } }

This is also denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E( g(X) \vert Y )} . Also, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Y} is supposed to be an event, not just a variable, that you are conditioning on (all of this is built up from set theory).

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(g(X) \vert Y=a) = \langle g(X) \vert Y=a \rangle}

Stochastic Processes

Sean's experience: intimidating topic, due to the approach of most professionals being one of rigor.

Want to dispel this: from what we've covered, we already know everything we need to know

Stochastic process - just like a random variable, but slightly more generalized

Random variable / trials / experiments:

We will allow an experimental trial to result in a family of outcomes, parameterized by a variable (universally denoted as t)

Now you can create a mapping, and the stochastic processes is the result of the mapping

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t) }

When we substitute in a random variable, e.g. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_1} , this becomes a random variable.

So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t)} is the family of outcomes, parameterized by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t}

Example: six random clock digits (e.g. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{7, 3, 6, 1, 2, 0}} ; we push a button, the numbers all change

• one approach: treat this as one large number
• another approach: each number is random, and can find a joint PDF between them
• stochastic process approach: assign a t value to each digit
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t = 0, 1, 2, 3, 4, 5} - one for each digit
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t=0)} is the family of outcomes for the first digit
  • Set theory: mapping from the digits that show up, to the random variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X}

Sample space: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S = { \mathbf{0, 1, 2, 3, 4, 5, 6, 7, 8, 9} }} (possible values that our clock digits can take on)

Random variable: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}}

The interesting fact is, there may be a sequence, so that they all depend on one another.

Shower Example

Water hits the wall of the shower, runs down in serpentines

Random paths that jump around along the wall

One experiment: a snapshot of the serpentine, and it's particular path.

The coordinate of length down the shower is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t}

The coordinate of horiz. displacement is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t)}

Can look at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{ X(t_n),X(t_m)}} , or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X(t_1) \vert X(t_0)}} , etc.

Important to maintain that we have a parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t} , because for a given experiment, there is (or may be) some connection between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t)} for all of the t's

e.g. for the serpentines - we know the path a serpentine takes follows some pattern (because it creates some curve)

Stock Market Example

Random variable: value of stock

Stock is only dependent on how much someone is willing to pay for it. But we can only know the price of the stock when there is a transaction.

So actually the stock price is a set of steps - a piecewise, not a continuous, function

Types of Stochastic Processes

A single realization of a stochastic process (a single realization of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t)} ) can be:

• Continuous/smooth
• Discontinuous
• Piecewise

Thoughts

Each value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t} can have a separate probability distribution

And we can have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t)} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Y(t)} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z(t)} , etc.

Recall: frequentist

If we had a perfect flow machine, and a perfect experimental instrument... and running a turbulent flow experiment

Turbulent experiment is a stochastic process

We run the experiment, and we measure the velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u,v,w} as a function of time.

Velocity is smooth and continuous, and may be related by some physics that we know

We run the experiment over and over and over again to get statistics.

Maybe we look at the velocity at another point...

Or maybe we look at it EVERYWHERE

So then we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x,y,z,t} as stochastic parameters - MULTIPLE stochastic process parameters

The value at a certain point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u(x,y,z,t)} is a random variable - parameterized on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x,y,z,t}

Distinction: ensemble average vs. time average

• When we run 10 experiments, by gathering velocity data at a certain point
• Averaging over time for a single experiment, is different from averaging over the 10 experimental values for a given t

U, V Velocity Example

Continuing with the flow experiment...

Looking at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{U(t_1),U(t_2)}} and their correlation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho_{U(t_1), U(t_2)}}

The correlation drops from 1 to 0 as time goes on

This is true of all turbulent velocity fields, due to the strong dependence on initial conditions

The correlation curve has to be continuous, since the velocity field is continuous and smooth in time.

Timescale it takes the correlation to go to zero: wikipedia:Lyapunov exponent (amount of time it takes for two curves to become completely uncorrelated)

Turbulence: tradeoff between strong correlation (due to direction-change by eddies), which will go from +1 to negative, and the decaying exponential

Markov Processes and Bernoulli Processes

We have a Bernoulli trial (outcome is 0 or 1), for a bunch of experiments.

The outcome Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t)} (where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t} is the experiment number) is either 0 or 1 - but it is like the clock example given earlier, because each of the experiments is independent:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X(t_1) \vert X(t_0)} = f_{X(t_1)}}

Markov process - the PDF at some time parameter, conditioned on all of the previous times, is only dependent on the previous time

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{ X(t_{n+1}) \vert X( t_{n} ), X( t_{n-1} ), X( t_{n-2} ), \dots } = f_{ X( t_{n+1} ) \vert X( t_{n} ) } }

Markov chain: each link in the chain is only connected to the previous link

Examples:

Card games: cards represent a memory of the past moves (NOT a Markov process)

Chutes and Ladders: where you end up next depends ONLY on your current location and what you roll on the dice

Following are some examples of subsets of Markov processes. Each person made a set of assumptions, and showed that the result had some cool properties. No information is given about what they proved... just some nomenclature so we're familiar with it.

Martingale Process

http://en.wikipedia.org/wiki/Martingale_%28betting_system%29

This is similar to a Bernoulli process, but it adds up - you add up the

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t+1) = X(t) + B}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} is a distributed Bernoulli trial, either -1 or 1, with some probability (0.5).

So if we take the expectation of B, it's 0 - it could be 1, or it could be -1.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle X(n+1) \rangle = X(n)}

Levy Process

Is a Markov proces...

Condition 1: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t=0) = 0}

Condition 2: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t_n) - X(t_{n-1})} are disjoint (independent), for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n}

Condition 3:

Example (to clarify)...

Wiener Process

wikipedia:Wiener process

Fits within Levy processes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X( t_{n+1} ) = X( t_n ) + \sqrt{ \delta^2 dt } N_{t}^{t + dt}(0,1) }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N_{t}^{t+1}} is a transition probability - transitioning from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t+dt}

This is an initial value problem (think explicit Euler...)

Can get the next value from the current value PLUS the RHS

C++ random number generator --> normally-distributed number, mean 0, standard deviation 1: that's Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N_{t}^{t+dt}} .

N is a value, but its value is distributed normally. A value comes out, we multiply it by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{\delta^2 dt}} , and add it to our current value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t_n)} to get the next value.

So this means our distribution will be discontinuous (due to random number).

Question: can we shrink Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle dt} small enough that our function is continuous?

For Euler method, this works fine:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{\Delta t \rightarrow 0} \frac{ X(t_{n+1}) - X(t_n) }{ \Delta t } = F(x) }

and we get a derivative,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ dX }{ dt } = F }

But now we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{\Delta t \rightarrow 0} \frac{ X(t_{n+1}) - X(t_n) }{ \Delta t } = \frac{1}{\sqrt{( \Delta t )} } F(x) }

and so as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t} goes to zero, the RHS goes to infinity

We can't do a derivative of this, so calculus becomes difficult on these stochastic processes

This one is not really continuous

But we still may be able to do calculus, because there's still an area under the curve

Probability: may use more generalized definitions of a limit

e.g. aren't looking at single realization values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} - if we're looking at the PDF at two points, we can say there is a limit in the PDF

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{\Delta t \rightarrow 0} \frac{ f_{X(t_{n+1})} - f_{X(t_n)} }{ \Delta t } }

There are four different probability limits that can be defined, and you can start doing calculus

Ito calculus, Strotanovich calculus, stochastic calculus, etc. - all go down the field of "not discrete, so need to define a new calculus"

But we're always dealing with smooth and continuous processes, so we won't go down that road

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{l} X(t=0) = 1 \\ X(t=1) = 1 + N \sim Normal(1,1) X(t=2) = X(t=1) + N \sim Normal + Normal \end{array} }

So you always have a normally-distributed process, but with increasing variance

Alternative definition of Markov process:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t_{n+1}) = X(t_n) + g( X(t), dt ) }

If your function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g} is normally distributed, with the same mean and variance, Central Limit Theorem tells you Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t)} will be normally distributed

Example of Wiener process:

Einstein's Brownian Motion Paper (1905)

3 papers:

• Brownian motion
• photoelectric effect
• special relativity

Pollen particle in water - randomly moving around, being bumped to the left, bumped to the right

Considered the first directly observable evidence of kinetic theory

http://files.charlesmartinreid.com/Einstein_BrownianMotion_1905.pdf

Ornstein-Uhlenbeck Process

Also a Levy process

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t_{n+1}) = X(t_n) - \gamma X(t_n) dt + \sqrt{ \beta^2 dt } N_{t}^{t+dt}(0,1) }

Second term on RHS is like an advection term

Third term is a random term like the one in the Wiener process

Example of Ornstein-Uhlenbeck process:

Langevin's Brownian Motion - let's let the position of a Brownian particle be goverened by Newton's second law

Have two variables: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V} , where the position Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} is continuous, and a velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V} which is a Wiener process

Langevin considered this to be more physically sensible - because particle position isn't discontinuous

Poisson Process

Like a Bernoulli process, but we're going to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t_{n+1}) = X(t_n) + 1 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t \sim Poisson( \lambda )}

The dt has a certain distribution - as opposed to the Wiener process, where we have an expression in terms of dt

So the y-value (variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} ) is uniformly distributed (well, uniform - everything is 1)

But the x-axis (variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t} ) jumps are Poisson-distributed

Examples:

• Radioactive decay
• Telephone calls arriving at a switchboard
• Page view request at web site

Things that, over a certain interval, are uniformly distributed

Random Walk Processes

Depending on the conditions on your steps, you end up with different types of random walks

e.g. if your steps are normally distributed, you have a Wiener random walk

if your steps are Bernoulli steps, it is a Martingale random walk

etc...

But ALL are Markov processes

October 28, 2010

(Missed some...)

Wiener process - distributed randomly about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_{0}}

We can get statistics from a Wiener process:

• Means Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu_{ X_n }} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu_{ X_{n+1} }}
• Variances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sigma_{X_n}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sigma_{ X_{n+1} }}
• Correlation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho_{X_n, X_{n+1}}} (how narrow the 2-D joint PDF surface is)

Wiener Process Statistics

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_{n+1} = X_{n} + \sqrt{ \delta^2 dt } N(0,1)}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N(0,1)} is the normal distribution with mean 0 and variance 1

This can also be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N(0, \sqrt{ \delta^2 dt } )}

Property of normal distribution (see wikipedia:Normal distribution) - if we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_1 \sim Normal(\mu_1,\sigma_1)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_2 \sim Normal(\mu_2,\sigma_2)} , then for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Y = a X_1 + b X_2 } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Y \sim N( a \mu_1 + b \mu_2, a^2 \sigma_1^2 + b^2 \sigma_2^2)}

Lemmons books:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \displaystyle{ \lim{at \leftarrow 0 } X_{n+1} = X_{n} }}

Which then leads to:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \displaystyle{ \lim{at \leftarrow 0 } \frac{ X_{n+1} - X_{n} }{dt} = \lim{ at \leftarrow 0 } \sqrt{ \frac{\delta^2}{dt} } N(0,1) }}

which is NOT SMOOTH!

Then we can say

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_{n+1} - X_{n} = ( X_{n+1} - X_{n+\frac{1}{2}} ) + ( X_{n+\frac{1}{2}} - X_{n} ) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{ \delta^2 dt } N_{t}^{t+1} (0,1) = \sqrt{ \delta^2 dt } N_{t+\frac{dt}{2}}^{t+dt} (0,1) + \sqrt{ \delta^2 dt } N_{t}^{t+\frac{dt}{2}} (0,1) }

all the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{ \delta^2 dt }} terms cancel out, and we get:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N(0,1) = N(0,1)}

which means the Wiener process is consistent (and independent of the dt selection)

Have a bunch of random processes, separated by intervals of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle dt} , and construct a PDF

Definition of stochastic process: Set of random variables... performing a trial of an experiment... outcome is a set of random variables parameterized by t

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t)}

When we put in a specific t (say Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_0} ), we plug it in and get a particular value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(t = t_0)}

So if we pick a value of t, we get a PDF for X at that t - e.g. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X(t_5)}} , or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X(t_3)}} , etc.

It would be great if we had the joint PDF, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{ X(t_5), X(t_3) }} , because then we can get the marginal PDF, the variances, the covariance, etc.

So what we've just shown is, no matter what our dt is, we always get the same PDF Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X(t_0)}} at a particular Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_0}

Wiener process is used more commonly because of this consistency

Analog for differential equations: if you have a differential equation, and you're integrating to a certain point in time, it doesn't matter if you take two timesteps or a thousand timesteps to get there, you will still get the same result.

Note: You can also get a similar consistency result for a Cauchy distribution, but because it doesn't have a finite variance, it isn't as useful

Now, using the method of moments,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(X_{n+1}) = E(X_n) + E \left( \sqrt{ \delta^2 dt } N_{n}^{n+1} (0,1) \right) }

We can take constants out of the expectation operator, so the last term becomes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} E \left( \sqrt{ \delta^2 dt } N_{n}^{n+1} (0,1) \right) &=& \sqrt{ \delta^2 dt } E \left( N_{n}^{n+1} (0,1) \right) \\ &=& 0 \end{array} }

Next, can write this as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt} E(X) = 0 }

or alternatively,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt} \langle X \rangle = 0 }

We have an ODE for the first moment.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \frac{d}{dt} Var(X) &=& \frac{d}{dt} \left( \langle X^2 \rangle - \langle X \rangle^2 \right) \\ &=& \frac{d \langle X^2 \rangle}{dt} - 2 \langle X \rangle \frac{ d \langle X \rangle }{ dt } \\ &=& \frac{d \langle X^2 \rangle}{dt} \end{array} }

Next,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} X_{n+1}^2 &=& \left[ X_{n} + \sqrt{ \delta^2 dt } N(0,1) \right]^2 \\ &=& X_{n}^2 + 2 \sqrt{ \delta^2 dt } X_n N_{t}^{t+dt} (0,1) + \delta^2 dt \left( N_t^{t+dt} (0,1) \right)^2 \end{array} }

And taking the expectation of both sides:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \langle X_{n+1}^2 \rangle &=& \langle X_{n}^2 \rangle + 2 \sqrt{ \delta^2 dt } \langle X_n N_{t}^{t+dt} (0,1) \rangle + \delta^2 dt \langle \left( N_t^{t+dt} (0,1) \right)^2 \rangle \end{array} }

For a Wiener process, the normal distribution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N_{t}^{t+dt}} is added after Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_n} , and is independent of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_n} (however, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_{n+1}} is not independent of the normal distribution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N_{t}^{dt}} , because that normal distribution is used to obtain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_{n+1}} )

So we can separate the expectation of these variables (because their joint DPF is equal to the product of their marginal PDFs), and the second expectation term becomes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle X_n N_{t}^{t+dt} \rangle = \langle X_n \rangle \times \langle N_{t}^{t+dt} \rangle }

and the second term goes to 0, so that term goes away.

In the third expectation term, the expectation of the squared normal distribution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle N_{t}^{t+dt} (0,1)^2 \rangle } is NOT a normal distribution - for example, the squared normal distribution can never be negative.

wikipedia:Chi-square distribution is the product of normal distributions... has mean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu=1} ... so the third expectation term goes to 1

Plugging in and simplifying yields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ d \langle X^2 \rangle }{dt} = \frac{d}{dt} Var(X) = \delta^2 }

So the variance increases at a constant rate.

Evolution of PDF in Time

Use successive substitution and same idea used before in finding that Wiener process independent of dt

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{x} (x;t) }

Inspection:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} X(dt) &=& X(0) + \sqrt{ \delta^2 dt } N_{0}^{dt} (0,1) \\ X(2 dt) &=& X(0) + N(0, \delta^2 (2 dt) ) \\ & \vdots & \\ X(t) &=& X(0) + N(0, \delta^2 t) \end{array} }

This just says that as we advance in time, the mean of the distribution doesn't change, and the variance is linear in time, with coefficient Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta^2} .

Einstein Paper

In Einstein's 1905 paper (see above for link), he finds the result:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ \partial f_{X} }{ \partial t } = \frac{ \delta^2 }{2} \frac{\delta^2 f_{X} }{ d x^2 }}

and he says this is the same as the heat equation, but for a distribution instead of a scalar:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x \sim \sqrt{ D t } }

The point being: there are three ways to get at the PDF:

• Method of moments (how the PDF evolves in time)
• Inspection (what we just did above)
• Einstein (the PDE of the PDF)

For the Wiener process, this is a lot easier than for other distributions - another 8 yrs for someone to do the same thing for the Wiener process with a drift term, e.g.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_{n+1} = X_{n} + \sqrt{ \delta^2 dt } N(0,1) + V dt }

This is still going to be a Markov process, but is no longer a Martingale process... Adding that one extra term made it extremely difficult to derive the equivalent governing equation PDE, and took 8 years

When it was derived, it was called the wikipedia:Fokker-Planck equation

Point: it's powerful when you can go from a stochastic differential equation (e.g. definition of Wiener process; difficult to deal with) to a PDE (easier to deal with)

To generalize the process and mathematics required to go from a general stochastic differential equation to a PDE requires wikipedia:Ito calculus, and we're not going to go there

Aside

Can take the joint PDF at two times,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X_1 , X_2 } }

and we can get the covariance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Cov(X_1 , X_2)} and the correlation function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho_{X_1, X_2}}

We can find an expression for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X_1, X_2}} and then use that to find the covariances and correlations.... or, we can find them directly from our stochastic differential equation

Covariance

Definition of covariance is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Cov(X,Y) = E( (X - \langle X \rangle ) ( Y - \langle Y \rangle ) ) }

so,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Cov(X_1, X_2) = E \left( (X_1 - \langle X_1 \rangle) (X_2 - \langle X_2 \rangle) \right) = E( X_1^{\prime} X_2^{\prime}) }

Now we can use the Wiener process stochastic differential equation to get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Cov(X_1, X_2) = E \left( (X_1^{\prime})^2 + X_{1}^{\prime} N( 0, \delta^2 dt ) \right) }

And we just did a couple of these operations above, so we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Cov(X_1, X_2) = \langle (X_1^{\prime})^2 \rangle = \delta^2 t }

Correlation

Definition of correlation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Corr(X_1, X_2) = \frac{ Cov( X_1, X_2 ) }{ \sqrt{ Var(X_1) Var(X_2) } }}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho_{X_1, X_2} = \frac{ Cov(X_1, X_2) }{ Var(X_1) Var(X_2) } = \frac{ \delta^2 t }{ \delta^2 t \times \delta^2 (t + dt) } = \sqrt{ \frac{1}{1 + \frac{dt}{t} } } }

When you plot this:

• Starts with a high correlation (starts at 1)
• It has a very long tail

Joint PDF: equal to the marginal PDF times the conditional PDF

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X_1, X_2} = f_{X_1} \times f_{X_2 \vert X_1 } }

What is the conditional PDF?

We know the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_1} at some time... what's the probability of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_2} given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_1} ?

A couple of observations:

The value of X_2 is governed by the stochastic differential equation (the one that defines the Wiener process)

And we know that the Wiener process has consistency... so regardless of the timestep we take, the Wiener process stochastic differential equation will still hold

Meaning the relationship is one of:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_{n+1} = X_{n} + \sqrt{ \delta^2 dt } N(0,1) }

and this means that X_2 is related to X_1 by just adding a normal distribution... and that distribution has a variance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta^2 t} and is added to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_1} (which meas it is a normal distribution with mean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_1} )

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{X_2 \vert X_1} = N(X_{1}, \delta^2 t) }

Continuous and Smooth Realizations of a Continuous-Time Stochastic Process

Want to show both of these identities:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle \frac{ \partial U }{ \partial t } \rangle = \frac{ \partial \langle U \rangle }{ \partial t } }

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle \frac{ \partial Q(U) }{ \partial t } \rangle = \int_{-\infty}^{\infty} Q(u) \frac{ \partial f_{U} (u; t) }{ \partial t } du }

This will make it straightforward to derive PDF transport equations.

For the first:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ \partial U }{ \partial t } = \lim_{t \leftarrow t^{\star} } \frac{ U(t^{\star}) - U(t) }{ t^{\star} - t } }

which becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \langle \frac{\partial U}{\partial t} \rangle &=& \langle \lim_{t \leftarrow t^{\star} } \frac{ U(t^{\star}) - U(t) }{ t^{\star} - t } \rangle = \lim_{t \leftarrow t^{\star} } \frac{ \langle U(t^{\star}) - U(t) \rangle }{ t^{\star} - t } \end{array} }

and the numerator becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \langle U(t^{\star}) - U(t) \rangle &=& \int_{-\infty}^{\infty} (u^{\star} - u) f_{U^{\star},U} (u; t^{\star},t) du^{\star} du \\ &=& \int_{-\infty}^{\infty} u^{\star} f_{U^{\star}} (u; t^{\star}) du^{\star} - \int_{-\infty}^{\infty} u f_{U} (u; t) du \\ &=& \langle U(t^{\star}) \rangle - \langle U(t) \rangle \end{array} }

For the second identity:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle \frac{ \partial Q(U) }{ \partial t } = \lim_{t \rightarrow t^{\star}} \frac{ \langle Q(U(t^{\star})) - Q(U(t)) \rangle }{ t^{\star} - t } }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle Q(U^{\star}) - Q(U) \rangle = \int \int_{\infty}^{\infty} \left( Q(u^{\star}) - Q(u) \right) f_{U^{\star},U} (u^{\star}, u; t^{\star}, t) du^{\star} du }

which becomes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle Q(U^{\star}) - Q(U) \rangle = \int_{-\infty}^{\infty} Q(u^{\star}) f_{U^{\star}}(u; t^{\star}) du - \int_{-\infty}^{\infty} Q(u) f_{U}(u; t) du }

(Note: removing the star from the dummy variable in the first integral, but still indicating a difference by including the star in the PDF variables)

Next step:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle Q(U^{\star}) - Q(U) \rangle = \int_{-\infty}^{\infty} Q(u) \left( f_{U^{\star}} (u; t^{\star}) - f_{U} (u; t) \right) du }

and bringing in the limit from above:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{t \rightarrow t^{\star} } \frac{ \langle Q(U(t^{\star})) - Q(U(t)) }{ t^{\star} - t } = \int_{-\infty}^{\infty} Q(u) \lim_{t^{\star} \rightarrow t} \frac{ f_{U^{\star}} (u; t^{\star}) - f_{U}(u; t) }{ t^{\star} - t} du }

which is equal to the final identity we were trying to prove in the first place...

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^{\infty} Q(u) \frac{ \partial f_{U}(u; t) }{ \partial t } du }

Derivation of PDF Transport Equation

Momentum equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho \frac{ DU_i }{Dt} = - \frac{ \partial p}{\partial x_i} - \frac{ \partial \tau_{ij} }{\partial x_j} + \rho g_i = A_i }

and species equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho \frac{ D \phi_{\alpha} }{ Dt } = - \frac{ \partial J_{j,\alpha} }{\partial x_j} + \rho S_{\alpha}(\boldsymbol{\phi}) = \Theta_{\alpha} }

Now we start out with a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q} , which has the following properties:

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q(u,\phi)}
• Scalar
• Only a function of one realization at one time (not Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U,\phi} at different times...)
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q} must be a nice function (can't go to infinity very fast, so that when we calculate the moments, the PDF times Q can't go to infinity - i.e. it doesn't go to infinity so fast it overpowers the tails of the PDF and therefore makes the moment integral infinity)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle \rho \frac{ D }{Dt} Q(U, \phi) \rangle = \langle \rho \frac{\partial Q}{\partial t} + \rho U_j \frac{ \partial Q }{ \partial x_j } \rangle = \langle \frac{ \partial \rho Q }{ \partial t } + \frac{ \partial \rho U_j Q }{ \partial x_j } \rangle = \frac{ \partial \langle \rho Q \rangle }{ \partial t } + \frac{ \partial \langle \rho U_j Q \rangle }{ \partial x_j } }

by the definition of the total derivative.

This can be expanded, using the definition of the expectation, as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle \rho \frac{ D }{Dt} Q(U, \phi) \rangle = \int \dots \int_{-\infty}^{\infty} Q(u, \psi) \left( \frac{ \partial \rho f_{U,\phi} }{ \partial t } + \frac{ \partial \rho u_j f_{U,\phi} }{ \partial x_j } \right) du d \psi }

Then, using the chain rule:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle \rho \frac{ DQ(U,\phi) }{ Dt } \rangle = \langle \rho \frac{\partial Q}{\partial U_i} \frac{D U_i}{D t} + \rho \frac{ \partial Q } {\partial \phi_{\alpha} } \frac{D \phi_{\alpha}}{Dt} \rangle = \langle \frac{\partial Q}{\partial U_i} A_i \rangle + \langle \frac{\partial Q}{\partial \phi_{\alpha}} \Theta_{\alpha} \rangle }

We are going to use the rule

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{U,\phi,z} = f_{U,\phi \vert z} f_{z} = f_{z \vert U,\phi} f_{U,\phi} }

to make

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle \rho \frac{ DQ(U,\phi) }{ Dt } \rangle = - \int \int_{-\infty}^{\infty} Q \frac{ \partial }{ \partial \psi_{\alpha}} \left( \langle \Theta_{\alpha} \vert u, \psi \rangle f_{U,\phi} \right) du d \psi - \int \int_{-\infty}^{\infty} Q \frac{ \partial }{ \partial u_j} \left( \langle A_{j} \vert u, \psi \rangle f_{U,\phi} \right) du d \psi }

So now we set these two equalities of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle \rho \frac{ DQ(U,\phi) }{ Dt } \rangle} equal to one another... and we're doing this for arbitrary Q... and as a result, we get the following:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ \partial \rho f_{U,\phi} }{ \partial t } + \frac{ \partial \rho u_j f_{U,\phi} }{\partial x_j} + \frac{ \partial }{\partial \psi_{\alpha}} \left( \langle \Theta_{\alpha} \vert u, \psi \rangle f_{U,\phi} \right) + \frac{ \partial }{\partial u_j } \left( \langle A_j \vert u, \psi \rangle f_{U,\phi} \right) = 0 }

Further Reading:

Don Lemmons, An Introduction to Stochastic Processes in Physics

• Very readable for undergraduates
• Not in-depth, covers all the basic concepts

Review Article (1943): Stochastic Problems in Physics and Astronomy. S. Chandrasekhar. Review of Modern Physics. Vol 15, p. 1-89.

• Much more in-depth
• But difficult - and

Gardiner, Handbook of Stochastic Methods.