06_17 - STAT 410 Examples for 06/17/2009 Summer 2009...

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Unformatted text preview: STAT 410 Examples for 06/17/2009 Summer 2009 Example 1 : Suppose a discrete random variable X has the following probability distribution: P( X = 0 ) = p , P( X = k ) = ! 2 1 k k , k = 1, 2, 3, a) Find the value of p that would make this a valid probability distribution. Must have = + 1 ! 2 1 k k k p = 1. Since a k k e k a ! = = , 1 ! 2 1 2 1 1- = = e k k k . Therefore, p + ( 1 2 1- e ) = 1 and p = 2 1 2 e- . b) Find E ( X ). E ( X ) = x x p x all ) ( = 0 ( 29 2 1 2 e- + = 1 ! 2 1 k k k k = ( 29 =- 1 ! 1 2 1 k k k = ( 29 =-- 1 1 ! 1 2 1 2 1 k k k = = ! 2 1 2 1 n n n = 2 2 1 e . c) Find the variance of X, Var ( X ). E ( X ( X 1 ) ) = ( 29 = - 2 ! 2 1 1 k k k k k = ( 29 =- 2 ! 2 2 1 k k k = ( 29 =-- 2 2 ! 2 2 1 4 1 k k k = = ! 2 1 4 1 n n n = 4 2 1 e . E ( X 2 ) = E ( X ( X 1 ) ) + E ( X ) = 2 1 4 3 e . Var ( X ) = E ( X 2 ) [ E ( X ) ] 2 = e e - 4 1 4 3 2 1 . d) Find the moment-generating function of X, M X ( t ). M X ( t ) = x x t x p e all ) ( = 1 ( 29 2 1 2 e- + = 1 ! 2 1 k k k t k e = ( 29 2 1 2 e- + = 1 ! 2 k k t k e = ( 29 2 1 2 e- + - 1 2 t e e = 2 2 1 1 t e e e +- . e) Use the moment-generating function of X, M X ( t ), to find E ( X ). ( 29 2 M 2 X ' t e e t e t = , E ( X ) = ( 29 2 M 2 1 X ' e = . f) Use the moment-generating function of X, M X ( t ), to find the variance of X, Var ( X ). ( 29 2 2 M 2 2 2 X ' ' t t e e e e t t e e t + = , E ( X 2 ) = ( 29 2 1 X 4 3 M ' ' e = . Var ( X ) = E ( X 2 ) [ E ( X ) ] 2 = e e - 4 1 4 3 2 1 . Example 2 : Suppose a random variable X has the following probability density function: =- otherwise 1 ) ( x C x f x e a) What must the value of C be so that f ( x ) is a probability density function? For f ( x ) to be a probability density function, we must have: 1) f ( x ) 0, 2) ( 29 1 d = - x x f . ( 29 -- - = = = 1 1 d d d 1 x C x C x x f x x e e ( 29 ( 29 - =- =- =-- e e e e C C C x 1 1 1 1 . Therefore, - = 1 e e C 1.5819767 . - =- otherwise 1 1 ) ( x x f x e e e b) Find the cumulative distribution function F ( x ) = P( X x )....
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06_17 - STAT 410 Examples for 06/17/2009 Summer 2009...

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