Midterm Solutions

Midterm Solutions - Stat 116 Midterm solution May 7 2008 1 Let X,Y be distributed independent exp and let 0< a< b Find P X> a | Y − X>

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Unformatted text preview: Stat 116 Midterm solution May 7, 2008 1. Let X,Y be distributed independent exp( λ ) and let 0 < a < b . Find P ( X > a | Y − X > b ). Solution: Using the definition of conditional expectation P ( X > a,Y − X > b ) = integraldisplay ∞ a integraldisplay ∞ x + b λ 2 e − λ ( x + y ) dydx = e − λ (2 a + b ) 2 also P ( Y − X > b ) = integraldisplay ∞ integraldisplay ∞ x + b λ 2 e − λ ( x + y ) dydx = e − λb 2 so P ( X > a | Y − X > b ) = P ( X > a,Y − X > b ) P ( Y − X > b ) = e − 2 λa . 1 2. Let X be a discrete r.v. taking values in 0 , 1 , 2 ,... . (a) Prove E X = ∑ ∞ k =0 P ( X > k ). ( Hint : substitute P ( X > k ) = ∑ ∞ i = k +1 P ( X = i ) in the right hand side) (b) Let T be the lifetime of a machine. Suppose the machine is working after t days, what is the expected subsequent lifetime when f T ( x ) = P ( T = x ) = ( N + 1) − 1 , x = 0 , 1 ,... ,N ? ( hint : Condition on the fact that the machine lasted until time t then use the result from part (a)) Solution: (a) summationdisplay k P ( X > k ) = ∞ summationdisplay k =0 ∞ summationdisplay i = k +1 P ( X = i ) = ∞ summationdisplay...
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This note was uploaded on 01/13/2010 for the course STATS 116 taught by Professor Staff during the Spring '07 term at Stanford.

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Midterm Solutions - Stat 116 Midterm solution May 7 2008 1 Let X,Y be distributed independent exp and let 0< a< b Find P X> a | Y − X>

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