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08_27 - STAT 410 Examples for expected value E X = X If...

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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 ) = ² ³ ² ´ µ < < 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 ) ) = ± - 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 ) ( x x p x - ° Var ( X ) = ( ) ± - - x x x d f ) ( 2 X ± = [ ] 2 2 ) X ( E ) ( x x x d f - · ¸ ¹ ¹ º » ± - Example 1 : x p ( x ) x 2 p ( x ) 1 0.2 0.2 2 0.4 1.6 3 0.3 2.7 4 0.1 1.6 E ( X 2 ) = 6.1 Var ( X ) = 6.1 – 2.3 2 = 0.81 6.1 Example 2 : f X ( x ) = ² ³ ² ´ µ < < o.w. 0
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