NotesSep22 - If(X Y is continuous with a joint pdf fX,Y...

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If ( X,Y ) is continuous with a joint pdf f X,Y then Cov( X,Y ) = Z -∞ Z -∞ ( x - μ X )( y - μ Y ) f X,Y ( x,y ) dxdy. If X and Y are discrete with possible values ( x i ) and ( y j ) and a joint pmf p X,Y , then Cov( X,Y ) = X x i X y j ( x i - μ X )( y j - μ Y ) p X,Y ( x i ,y j ) .
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An alternative formula for covarance: Cov( X,Y ) = E ( XY ) - ( EX )( EY ) = E ( XY ) - μ X μ Y .
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If ( X,Y ) is continuous with a joint pdf f X,Y then Cov( X,Y ) = E ( XY ) - μ X μ Y = Z -∞ Z -∞ xy f X,Y ( x,y ) dxdy - μ X μ Y , If X and Y are discrete with possible values ( x i ) and ( y j ) and a joint pmf p X,Y , then Cov( X,Y ) = E ( XY ) - μ X μ Y = X x i X y j x i y j p X,Y ( x i ,y j ) - μ X μ Y .
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Example Let ( X,Y ) be a continuous random vector with a joint pdf f X,Y ( x,y ) = 15 xy 2 , 0 x,y 1 , y x. Compute the covariance of X and Y .
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Intuition Suppose that random variables X and Y are related in such a way that if X is above its mean μ X then Y also tends to be above its mean μ Y , and other way around. Then the product ( X - μ X )( Y - μ Y ) tends to be positive, and so the covariance of X and Y , Cov( X,Y ) = E [( X - μ X )( Y - μ Y )] , will be positive as well.
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Suppose that if X is above its mean μ X then Y tends to be below its mean μ Y , and if X is below its mean μ X then Y tends to be above its mean μ Y . Then the product (
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NotesSep22 - If(X Y is continuous with a joint pdf fX,Y...

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