Statistical Treatment of Experimental Data

"Statistical Treatment of Experimental Data" by Green and Margerison (Elsevier)

Chapter 2 - Probability

Basic definitions:

set of all possible outcomes from random experiment - sample space
discrete - countable number of possible outcomes (can also be infinite - as in, number of particles emitted)
continuous - all possible real values in certain interval or series of intervals may occur
univariate - only one number is recorded
multivariate -more than one value obtained from single performance of an experiment
event - set of outcomes in the sample space
probability of an event A as outcome is P(A)
addition law: P(A U B) = P(A) + P(B)
venn diagram: if two events are not mutually exclusive, split into three mutually exclusive events (D - (D and E)), (E - (D and E)), (D and E)
product law: P(A and B) = P(A) * P(B)
conditional probability: P(C | D) = P(C and D)/P(D)
independent - two or more performances of an experiment are called independent if probabilities of different outcomes in one are unaffected by outcomes in the other
replicates - independent repeat performances of an experiment

Probability models:

discrete uniform model - each outcome equally likely (e.g., tossing unbiased fair die)
random sampling - drawing random sample of size s from batch of size N (random means, all samples of size s equally likely to be chosen); number of possible samples is N choose s

$C(N,s) = \dfrac{N!}{s! (N-s)!}$

if r of the N items are special, number of ways of drawing sample containing d specials (number of ways of choosing d specials and s-d non-specials) is:

$C(r,d) \times C(N-r, s-d)$

$P(d\mbox{ specials}) = \dfrac{ C(r,d) C(N-r, s-d) }{ C(N,s) } \qquad d = 1, 2, ..., \min(r,s)$

this is the definition of hypergeometric distribution (special case of the uniform model)
example: bag with 3 red and 4 blue discs, no replacement; random sample of size 2 (=s) from batch of size 7 (=N) with 3 (=r) special (red). probability that 1 (=d) sample is special (red), is P(R and B) = 3 choose 1 * 4 choose 1 / 7 choose 2