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lecture11

lecture11 - Stat 302 Introduction to Probability Jiahua...

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Unformatted text preview: Stat 302, Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 11 January-April 2011 1 / 21 Expectation of g ( X , Y ) Suppose X and Y are jointly (absolutely) continuous with joint pdf given by f ( x , y ) . Let g ( x , y ) be a bi-variate function. Then we have E { g ( X , Y ) } = integraldisplay integraldisplay R 2 g ( x , y ) f ( x , y ) dxdy = integraldisplay integraldisplay R 2 g ( s , t ) f ( s , t ) dsdt . Jiahua Chen () Lecture 11 January-April 2011 2 / 21 Expectation of g ( X , Y ) Suppose X and Y are jointly (absolutely) continuous with joint pdf given by f ( x , y ) . Let g ( x , y ) be a bi-variate function. Then we have E { g ( X , Y ) } = integraldisplay integraldisplay R 2 g ( x , y ) f ( x , y ) dxdy = integraldisplay integraldisplay R 2 g ( s , t ) f ( s , t ) dsdt . We assume the integration converges absolutely. Jiahua Chen () Lecture 11 January-April 2011 2 / 21 Example: Expectation of g ( X , Y ) Suppose X and Y have joint pdf f ( x , y ) = 2 exp ( − x − y ) for 0 < x < y < ∞ . Let g ( x , y ) = ( y − x ) 2 . Then we have E { g ( X , Y ) } = 2 integraldisplay ∞ x = bracketleftbigg integraldisplay ∞ y = x ( y − x ) 2 exp ( − ( y − x ) − 2 x ) dy bracketrightbigg dx = 2 integraldisplay ∞ x = exp ( − 2 x ) bracketleftbigg integraldisplay ∞ u = u 2 exp ( − u ) du bracketrightbigg dx = 2 Γ ( 3 ) integraldisplay ∞ x = exp ( − 2 x ) dx = 2. Jiahua Chen () Lecture 11 January-April 2011 3 / 21 Example: Expectation of g ( X , Y ) Suppose X and Y have joint pdf f ( x , y ) = 2 exp ( − x − y ) for 0 < x < y < ∞ . How much is var ( X − Y ) ? Note that E { ( Y − X ) } = 2 integraldisplay ∞ x = bracketleftbigg integraldisplay ∞ y = x ( y − x ) exp ( − ( y − x ) − 2 x ) dy bracketrightbigg dx = 2 integraldisplay ∞ x = exp ( − 2 x ) bracketleftbigg integraldisplay ∞ u = u exp ( − u ) du bracketrightbigg dx = 2 Γ ( 2 ) integraldisplay ∞ x = exp ( − 2 x ) dx = 1. Therefore, var ( Y − X ) = E ( Y − X ) 2 − { E ( Y − X ) } 2 = 1. Jiahua Chen () Lecture 11 January-April 2011 4 / 21 Independence of X and Y Let X and Y be two random variables....
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