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04rand - EECS 501 RANDOM VARIABLES Fall 2001 So far Used...

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EECS 501 RANDOM VARIABLES Fall 2001 So far: Used sample space Ω and event space A to describe outcome. Now: Use a number to describe the outcome of an experiment. DEF: A random variable x is a mapping x : Ω → R (Ω=sample space). x associates a number with each outcome of an experiment. EX: Flip coin 100 times. x = (# heads ) 3 ; y =#heads in 1 st 10 flips. Note: Numbers represent outcomes, so sets of numbers represent events. Q: For what subsets F ⊂ R can we compute Pr [ x F ]? A1: Induced prob. space: Pr [ x F ] = Pr [ ω Ω : x ( ω ) F ] = Pr [ x - 1 ( F )]. Probability space (Ω , A , Pr ) and random variable x define ( preimage ) induced probability space ( R , A x , P x ) where P x [ F ] = Pr [ x - 1 ( F )]. EX: Suppose F ∈ A x P x [ F ] = 0 or 1. Prove x=constant with prob. 1. Huh? Heuristically, each sample point ω of Ω maps to same constant c . Proof: 1 = Pr [Ω] = Pr [ x - 1 ( R )] = Pr [ x - 1 ( n = -∞ [ n, n + 1))] = n = -∞ Pr [ x - 1 ([ n, n + 1))] = n = -∞ P x [[ n, n + 1)]. But: Given that P x [[ n, n + 1)] = 0 or 1. So P x [[ N, N + 1)] = 1 for some N .
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