Elements of Statistical Methods Probability (Ch 3) - Statistics

Finite Sample SpacesSuppose S = {s 1 ,...,s N } is a finite sample space with outcomes s 1 ,...,s N .Assume that C is the collection of all subsets (events) of S.and that we have a probability measure P defined for all A ∈ C .Denote by p i = P({s i }) the probability of the event {s i }.Then for any event A consisting of outcomes s i1 ,...,s ik we have⎛ ⎞ ⎛ ⎞[P(A) = P⎝ k {s i j} ⎠ = P⎝ [{s i } ⎠ = ∑ P({s i }) = ∑ p i (1)s i ∈As i ∈Aj=1s i ∈AThe probabilities of the individual outcomes determine the probability of any event.To specify P on C , we only need to specify p 1 ,..., p N with 0 ≤ p i ,i = 1,...,Nand p 1 + ... + p N = 1.This together with (1) defines a probability measure on C , satisfying axioms 1-3.This also works for denumerable S with 0 ≤ p i ,i = 1,2,... and ∑ ∞ i=1 p i = 1.13