slides22 - Lecture Stat 302 Introduction to Probability -...

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Lecture Stat 302 Introduction to Probability - Slides 22 AD April 2010 AD () April 2010 1 / 11
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Characterizing Joint Distributions/Densities: Covariance Consider two r.v. X and Y (either discrete or continuous), then the covariance of ( X , Y ) is de&ned as Cov ( X , Y ) = E (( X & E ( X )) ( Y & E ( Y ))) = E ( XY ) & E ( X ) E ( Y ) The covariance measures the degree to which X and Y vary together. If the covariance is positive, X tends to be larger than its mean when Y is larger than its mean. The covariance of a variable with itself is the variance of that variable. AD () April 2010 2 / 11
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Independent Variables and Covariance If X and Y are two independent r.v. then Cov ( X , Y ) = 0 Proof. We are going to show that E ( XY ) = E ( X ) E ( Y ) if X and Y are independent E ( XY ) = Z Z xy & f ( x , y ) dxdy = Z Z xy & f X ( x ) f Y ( y ) dxdy (independence) = & Z x & f X ( x ) dx ±& Z y & f Y ( y ) dy ± = E ( X ) E ( Y ) AD () April 2010 3 / 11
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Example: Two Stocks Let X and Y denote the values of two stocks at the end of a &ve-year period. X is uniformly distributed on ( 0 , 12 ) . Given X = x , Y is uniformly distributed on the interval ( 0 , x ) . Determine Cov ( X , Y ) . We have for 0 < x < 12 and 0 < y < x f ( x , y ) = f X ( x ) f Y j X ( y j x ) = 1 12 1 x so E ( X ) = Z 12 0 x & 1 12 dx = 6 , E ( Y ) = Z 12 0 Z x 0 y & 1 12 1 x dydx = 3 , E ( XY ) = Z 12 0 Z x 0 xy & 1 12 1 x dydx = 24 .
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slides22 - Lecture Stat 302 Introduction to Probability -...

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