6.8 Two Functions of Two Random Variables
Let X and Y be two random variables with a given joint PDF fXY(x, y). Assume that U and W are two functions of X and Y; that is, U = g(X, Y) and W = h(X, Y). Sometimes it is necessary to obtain the joint PDF of U and W, fUW(u, w), in terms of the PDFs of X and Y.
It can be shown that if (x1, y1), (x2, y2), …, (xn, yn) are the real solutions to the equations u = g(x, y) and w = h(x, y) then fUW(u, w) is given by
(6.15)
where J(x, y) is called the Jacobian of the transformation {u = g(x, y), w = h(x, y)} and is defined by
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