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Unformatted text preview: Stat 116 Midterm solution May 7, 2008 1. Let X,Y be distributed independent exp( Î» ) and let 0 < a < b . Find P ( X > a | Y âˆ’ X > b ). Solution: Using the definition of conditional expectation P ( X > a,Y âˆ’ X > b ) = integraldisplay âˆž a integraldisplay âˆž x + b Î» 2 e âˆ’ Î» ( x + y ) dydx = e âˆ’ Î» (2 a + b ) 2 also P ( Y âˆ’ X > b ) = integraldisplay âˆž integraldisplay âˆž x + b Î» 2 e âˆ’ Î» ( x + y ) dydx = e âˆ’ Î»b 2 so P ( X > a | Y âˆ’ X > b ) = P ( X > a,Y âˆ’ X > b ) P ( Y âˆ’ X > b ) = e âˆ’ 2 Î»a . 1 2. Let X be a discrete r.v. taking values in 0 , 1 , 2 ,... . (a) Prove E X = âˆ‘ âˆž k =0 P ( X > k ). ( Hint : substitute P ( X > k ) = âˆ‘ âˆž 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) âˆ’ 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 ) = âˆž summationdisplay k =0 âˆž summationdisplay i = k +1 P ( X = i ) = âˆž 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.
- Spring '07