01_18 - STAT 410 Examples for Spring 2008 Example 4 Suppose...

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Unformatted text preview: STAT 410 Examples for 01/18/2008 Spring 2008 Example 4 : Suppose a discrete random variable X has the following probability distribution: P( X = 0 ) = 2 1 2 e- , P( X = k ) = ! 2 1 k k ⋅ , k = 1, 2, 3, … d) Find the moment-generating function of X, M X ( t ). M X ( t ) = & ⋅ x x t x p e all ) ( = 1 ⋅ ( ) 2 1 2 e- + & ∞ = ⋅ ⋅ 1 ! 2 1 k k k t k e = ( ) 2 1 2 e- + & ∞ = ¡ ¡ ¢ £ ¤ ¤ ¥ ¦ 1 ! 2 k k t k e = ( ) 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 ). ( ) 2 M 2 X ' t e e t e t ⋅ = , E ( X ) = ( ) 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 ). ( ) 2 2 M 2 2 2 X ' ' t t e e e e t t e e t ⋅ ⋅ + ¡ ¡ ¢ £ ¤ ¤ ¥ ¦ = , E ( X 2 ) = ( ) 2 1 X 4 3 M ' ' e ⋅ = . Var ( X ) = E ( X 2 ) – [ E ( X ) ] 2 = e e ⋅ ⋅- 4 1 4 3 2 1 . Example 5 : Let X be a discrete Binomial ( n , p ) random variable. That is, suppose the ) random variable....
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01_18 - STAT 410 Examples for Spring 2008 Example 4 Suppose...

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