01_14 - x x p x all If ´ ∞ ∞-⋅ x x f x d< ∞ E X...

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STAT 410 Examples for 01/14/2008 Spring 2008 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 ) 1 0.2 0.2 2 0.4 0.6 3 0.3 0.9 4 0.1 1.0 F ( x ) = ² ² ² ³ ² ² ² ´ µ < < < < 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 ) = ± ² ³ < < 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. ----------------------------------------------------------------------------------------------------------------- expected value E ( X ) = μ X discrete continuous If µ x x p x all ) ( < , E ( X ) = µ
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Unformatted text preview: 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 ) = & ± & ² ³ < < o.w. 1 3 2 x x E ( X ) = μ X = ´ ⋅ 1 2 3 x x x d = ´ 1 3 3 x x d = 4 3 = 0.75. Example 3 : ( 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 = ∞ ....
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01_14 - x x p x all If ´ ∞ ∞-⋅ x x f x d< ∞ E X...

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