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Lecture

Probability spaces and random variables.pdf

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Department
Statistics
Course
STAT312
Professor
Douglas Wiens
Semester
Fall

Description
85 16. Probability spaces and random variables • We start with a sample space , whose elements are all possible outcomes of an experiment (e.g. toss a coin three times, is all possible sequences of three s and s). A Borel eld or -algebra of events is a collectioB of subsets (‘events’) of such that one of its elements is itself, it is closed under complementation, and closed under the taking of countable unions. • A probability is a function dened on B such that ( ) = 1 0 ( ) 1, and probabili- ties of disjoint countable unions are additive. The triple ( B ) is called a probability space. All the usual rules for manipulating probabilities fol- low from these axioms. e.g. ( ) = 1 ( ), ( ) = 0, ( ) ( ) if . • A (real valued, nite) random variable (r.v.) is a function : R with the property that 86 if is any open subset ofR , then 1( ) = { | ( ) } is an event, i.e. a member B. E.g. ( ) = # of heads in the sequence of tosses. (For a nite sample space we generally take B to be the set of all subsets .) n o — Note that 1( ) = 1( ) : 1 ( ) = { | ( ) } = { | ( ) } = {n| ( )o } = ( ) — By the preceding points, if is nlosed thon = is open and so 1( ) = 1( ) B : the inverse images of closed sets must also be events. • Since the set = ( ] is closed, so also 1( ) = { | ( ) } is a member of B, hence has a probability. We write ( ) = ({ | ( ) }) = ( ) 87 and call the distribution function (d.f.) of the
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