09_22ans - STAT 400 0. Fall 2011 Examples for 09/22/2011...

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STAT 400 Examples for 09/22/2011 Fall 2011 0. Let X be a random variable distributed uniformly over the interval [ a , b ]. Find the moment-generating function of X. M X ( t ) = E ( e t X ) = ( ) - dx x f x t e = - b a x t dx a b e 1 = a b t a b x t e 1 - = ( ) a b t a t b t e e - - , t 0. M X ( 0 ) = 1. 0.25. Find the moment generating function of an exponential random variable. M X ( t ) = E ( e t X ) = ( ) - dx x f x t e = - 0 θ 1 θ dx x x t e e = ( ) - 0 1 θ 1 θ dx t x e = ( ) 0 θ θ 1 θ 1 1 - - t t x e = θ 1 1 t - , t < 1 / θ . OR M X ( t ) = E ( e t X ) = ( ) - dx x f x t e = - 0 λ λ dx x x t e e = ( ) - 0 λ λ dx t x e = ( ) 0 λ λ λ - - t t x e = t λ λ - , t < λ .
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Suppose X has Exponential distribution with mean θ . f X ( x ) = θ 1 θ x e - , x > 0. a) Find P ( X > t ) for t > 0. P ( X > t ) = - t x dx e 1 θ θ = t x e - - θ = θ t e - , t > 0. b) Show that for positive t and s , P ( X > t + s | X > t ) = P ( X > s ) ( memoryless property ). P ( X > t + s | X > t ) = ( ) ( ) X P X X P t t s t > > + > = ( ) ( ) X P X P t s t > + > = ( ) θ θ t s t e e - + - = θ s e - = P ( X > s ). 0.75. Suppose X has Geometric distribution with probability of “success” p . a) Find P ( X > a ) for a = 0, 1, 2, 3, … . For Geometric ( p ), P ( X > a ) = ( ) - + = - 1 1 1 a k k p p = ( ) ( ) 1 1 1 p p p a - - - = ( 1 – p ) a , a = 0, 1, 2, 3, … . OR For Geometric ( p ), X = number of independent attempts needed to get the first “success”. P
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09_22ans - STAT 400 0. Fall 2011 Examples for 09/22/2011...

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