08_27 - STAT 410 Examples for 08/27/2008 Fall 2008 expected...

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Unformatted text preview: STAT 410 Examples for 08/27/2008 Fall 2008 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. 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 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 = ∞ . ----------------------------------------------------------------------------------------------------------------- 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 ) ) = ¡ ∞ ∞- ⋅ x x f x g d ) ( ) ( variance Var ( X ) = 2 X & = E ( [ X - μ X ] 2 ) = E ( X 2 ) – [ E ( X ) ] 2 discrete continuous Var ( X ) = ( ) & ⋅- x x p x all 2 X ) ( ¡ = [ ] 2 all 2 ) X ( E )...
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This note was uploaded on 10/22/2009 for the course STAT 410 taught by Professor Alexeistepanov during the Fall '08 term at University of Illinois at Urbana–Champaign.

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08_27 - STAT 410 Examples for 08/27/2008 Fall 2008 expected...

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