stat 410 - X x = 1 f X x F X x Example 3 f X x =>-o.w 1 5...

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STAT 410 Examples for 08/22/2011 Fall 2011 random variables discrete continuous probability mass function p.m.f. p ( x ) = P ( X = x ) probability density function p.d.f. f ( x ) x 0 p ( x ) 1 ( ) x x p all = 1 x f ( x ) 0 ( ) - x x f d = 1 cumulative distribution function c.d.f. F ( x ) = P ( X x ) ( ) x y y p ( ) - x d y y f Example 1: x p ( x ) F ( x ) F ( x ) = < < < < 4 1 4 3 9 . 0 3 2 6 . 0 2 1 2 . 0 1 0 x x x x x 1 0.2 0.2 2 0.4 0.6 3 0.3 0.9 4 0.1 1.0
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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
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Unformatted text preview: X ( x ) = 1. f X ( x ) F X ( x ) Example 3: f X ( x ) = >-o.w. 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 ) Example 4: ( Standard ) Cauchy distribution: f X ( x ) = ( ) 2 1 1 x + π , – ∞ < x < ∞ . F X ( x ) = ( ) ∫ ∞ +-x dy y 2 1 1 = ( ) 2 1 1 arctan + x , – ∞ < x < ∞ ....
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stat 410 - X x = 1 f X x F X x Example 3 f X x =>-o.w 1 5...

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