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

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Unformatted text preview: EECS 501 RANDOM VARIABLES Fall 2001 So far: Used sample space 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 : R (=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 [ : x ( ) F ] = Pr [ x- 1 ( F )]. Probability space ( , 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 of maps to same constant c . Proof: 1 = Pr [] = Pr [ x- 1 ( R )] = Pr [ x- 1 ( n =- [ n, n + 1))] = n =- Pr [ x...
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This note was uploaded on 07/22/2011 for the course EECS 370 taught by Professor Lehman during the Spring '10 term at University of Florida.

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

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