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chap4slidespt3 - Product of experiments Introduction Some...

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Product of experiments Introduction Some random experiments may be viewed as a compounded experiment: made up of “smaller”, sometimes independent, experiments that occur sequentially and/or concurrently in time. Examples include: Flipping two coins; Flipping a coin and rolling a die, etc. ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 56
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Product of experiments Basic model Definition We say that the random experiment ( S, F ,P ) is the product of n random experiments (trials) ( S i , F i ,P i ) if S = S 1 × S 2 × ··· × S n , F is the smallest σ -algebra containing all cartesian products of the type A 1 × A 2 × ... × A n , with A i ∈ F i . For any A i ∈ F i , i = 1 ,...,n , we have P ( S 1 × ··· × S i - 1 × A i × S i +1 × ··· × S n ) = P i ( A i ) ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 57
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Product of experiments Remark Let A i ∈ F i , i = 1 ,...,n : Event S 1 × ··· × S i - 1 × A i × S i +1 × ··· × S n occurs if and only A i occurs at the i th experiment ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 58
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Product of experiments Example Random experiment, sequence of two sub-experiments: 1 ( S 1 , F 1 ,P 1 ) : Flip a fair coin Sample space: S 1 = { H,T } Events algebra: F 1 = P S 1 Probability function: P 1 ( { H } ) = P 1 ( { T } ) = 1 / 2 2 ( S 2 , F 2 ,P 2 ) : Flip a fair coin Sample space: S 2 = { H,T } Events algebra: F 2 = P S 2 Probability function: P 2 ( { H } ) = P 2 ( { T } ) = 1 / 2 ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne) 59
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Product of experiments Example ( S, F ,P ) : Product experiment Sample space: S = S 1 × S 2 = { HH,HT,TH,TT } Events algebra: F = P S 1 × S 2 Probability function: P ( { HH } ) = 3 / 8 P ( { HT } ) = 1 / 8 P ( { TH } ) = 1 / 8 P ( { TT } ) = 3 / 8 We have: P ( S 1 × { T } ) = P ( { HT,TT } ) = 4 /
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chap4slidespt3 - Product of experiments Introduction Some...

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