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# 06_15 - STAT 410 Examples for random variables Summer 2009...

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STAT 410 Examples for 06/15/2009 Summer 2009 random variables discrete continuous probability mass function p.m.f. p ( x ) = P ( X = x ) probability density function p.d.f. f ( x ) 2200 x 0 p ( x ) 1 ( x x p all = 1 2200 x f ( x ) 0 ( 29 - x x f d = 1 cumulative distribution function c.d.f. F ( x ) = P ( X x ) ( x y y p ( 29 - x d y y f Example 1 : x p ( x ) F ( x ) 1 0.2 0.2 2 0.4 0.6 3 0.3 0.9 4 0.1 1.0 F ( x ) = < < < < 4 1 4 3 9 . 0 3 2 6 . 0 2 1 2 . 0 1 0 x x x x x

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Example 2 : f X ( x ) = < < o.w. 0 1 0 3 2 x x x < 0 F X ( x ) = 0. 0 x < 1 F X ( x ) = x d y y 0 2 3 = x 3 . x 1 F X ( x ) = 1. f X ( x ) F X ( x ) Example 3 : f X ( x ) = - o.w. 0 1 5 6 x x x < 1 F X ( x ) = 0. x 1 F X ( x ) = - x d y y 1 6 5 = – 1 5 x y - = 1 – x 5 . f X ( x ) F X ( x )
expected value E ( X ) = μ X discrete continuous If x x p x all ) ( < , E ( X ) = x x p x all ) ( If - x x f x d ) ( < , E ( X ) = - x x f x d ) ( Example 1 : x p ( x ) x p ( x ) 1 0.2 0.2 2 0.4 0.8 3 0.3 0.9 4 0.1 0.4 E ( X ) = μ X = 2.3. 2.3 Example 2 : f X ( x ) = < < o.w. 0 1 0 3 2 x x E ( X ) = μ X = 1 0 2 3 x x x d = 1 0 3 3 x x d = 4 3 = 0.75. Example 4 : ( Standard ) Cauchy distribution: f X ( x ) = ( 2 1 1 x + π , < x < . Even though f X ( x ) is symmetric about zero, E ( X ) is undefined since ( + - dx x x 1 1 2 π = .

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----------------------------------------------------------------------------------------------------------------- discrete continuous If x x p x g all ) ( ) ( < , E ( g ( X ) ) = x x p x g all ) ( ) ( If - x x f x g d ) ( ) ( < , E ( g ( X ) ) =
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06_15 - STAT 410 Examples for random variables Summer 2009...

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