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chap5slides - Chapter 5 Introduction to Random Variables...

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Chapter 5 Introduction to Random Variables ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 1

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Random Variables Introduction In many random experiments: We are interested in numerical quantities derived from the experimental outcomes, or It is convenient to assign numerical labels to outcomes. ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 2
Random Variables Introduction Example Experiment: A fair coin is flipped twice. Sample space: S = { HH, HT, TH, TT } Events algebra: F = P S = {∅ , { HH } , . . . , S } We are interested in the number of tails ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 3

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Random Variables Introduction Example Assigning labels S HH HT TH TT 0 1 2 R ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 4
Random Variables Introduction Example Experiment: Pick a car out of a black Escort ( Eb ), a red Escort ( Er ) and a yellow Mazda ( My ). Sample space: S = { Eb, Er, My } Events algebra: F = P S We are only interested in the make of the car not the color. ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 5

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Random Variables Introduction Example Assigning labels S Eb Er My 0 1 R ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 6
Random Variables Introduction Mathematically speaking... Sample space: S . We define a function from S into R X : s S X ( s ) R . ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 7

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Random Variables Definition Definition Let ( S, F , P ) be a probability space. A function X : s S X ( s ) R is called a random variable (RV) if for all x R : { s S : X ( s ) x } ∈ F . ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 8
Random Variables Introduction S s 1 s 2 s 3 s 4 s n X ( s 1 ) X ( s 2 ) X ( s 3 ) X ( s 4 ) = X ( s n ) R Domain of X ( · ) Range of X ( · ) : R X = { X ( s ) : s S } ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 9

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Random Variables Remarks The function X ( · ) is called a Random Variable (RV), because: The value X ( s ) depends on the outcome s .
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