09_04ans - STAT 410 Examples for 09/04/2009 Fall 2009 Mixed...

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STAT 410 Examples for 09/04/2009 Fall 2009 Mixed Random Variables : 1. Consider a random variable X with c.d.f. F ( x ) = < + - < 2 1 2 1 4 2 2 1 0 2 x x x x x a) Find μ X = E ( X ). b) Find σ X 2 = Var ( X ). Discrete portion of the probability distribution of X: p ( 1 ) = 1 / 4 , p ( 2 ) = 1 / 2 . Continuous portion of the probability distribution of X: f ( x ) = < < - o.w. 0 2 1 2 1 x x . a) μ = E ( X ) = 1 1 / 4 + 2 1 / 2 + - 2 1 2 1 x x x d = 5 / 3 . b) E ( X 2 ) = 1 2 1 / 4 + 2 2 1 / 2 + - 2 1 2 2 1 x x x d = 71 / 24 . σ 2 = Var ( X ) = E ( X 2 ) [ E ( X ) ] 2 = 13 / 72 .
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1.9.20 Let X be a random variable of the discrete type with pmf p ( x ) that is positive on the nonnegative integers and is equal to zero elsewhere. Show that E ( X ) = ( 29 [ ] - = 0 F 1 x x , where F ( x ) is the cdf of X. Idea
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09_04ans - STAT 410 Examples for 09/04/2009 Fall 2009 Mixed...

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