6.8 Two Functions of Two Random Variables

Let X and Y be two random variables with a given joint PDF fXY(xy). Assume that U and W are two functions of X and Y; that is, U = g(XY) and W = h(XY). Sometimes it is necessary to obtain the joint PDF of U and W, fUW(uw), in terms of the PDFs of X and Y.

It can be shown that if (x1y1), (x2y2), …, (xnyn) are the real solutions to the equations u = g(xy) and w = h(xy) then fUW(uw) is given by

fUWuw=fXYx1y1Jx1y1+fXYx2y2Jx2y2++fXYxnynJxnyn

si106_e  (6.15)

where J(xy) is called the Jacobian of the transformation {u = g(xy), w = h(xy)} and is defined by

Jxy=gxgyhxhy=gxhy

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