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elemprob-page20 - The multivariate distribution function...

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The multivariate distribution function of ( X, Y ) is defined by F X,Y ( x, y ) = P ( X x, Y y ). In the continuous case, this is x -∞ y -∞ f X,Y ( x, y ) dy dx, and so we have f ( x, y ) = 2 F ∂x∂y ( x, y ) . The extension to n random variables is exactly similar. We have P ( a X b, c Y d ) = b a d c f X,Y ( x, y ) dy dx, or P (( X, Y ) D ) = D f X,Y dy dx when D is the set { ( x, y ) : a x b, c y d } . One can show this holds when D is any set. For example, P ( X < Y ) = { x<y } f X,Y ( x, y ) dy dx. If one has the joint density of X and Y , one can recover the densities of X and of Y : f X ( x ) = -∞ f X,Y ( x, y ) dy, f Y ( y ) = -∞ f X,Y ( x, y ) dx. A multivariate random vector is ( X 1 , . . . , X r ) with P ( X 1 = n 1 , . . . , X r = n r ) = n ! n 1 ! · · · n r ! p n 1 1 · · · p n r r , where n 1 + · · · + n r = n and p 1 + · · · p r = 1. In the discrete case we say X and Y are independent if P ( X = x, Y = y ) = P ( X = x, Y = y ) for all x and y . In the continuous case,
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