## 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

${\mathit{f}}_{\mathit{UW}}\left(\mathit{u},\mathit{w}\right)=\frac{{\mathit{f}}_{\mathit{XY}}\left({\mathit{x}}_{1},{\mathit{y}}_{1}\right)}{\left|\mathit{J}\left({\mathit{x}}_{1},{\mathit{y}}_{1}\right)\right|}+\frac{{\mathit{f}}_{\mathit{XY}}\left({\mathit{x}}_{2},{\mathit{y}}_{2}\right)}{\left|\mathit{J}\left({\mathit{x}}_{2},{\mathit{y}}_{2}\right)\right|}+\dots +\frac{{\mathit{f}}_{\mathit{XY}}\left({\mathit{x}}_{\mathit{n}},{\mathit{y}}_{\mathit{n}}\right)}{\left|\mathit{J}\left({\mathit{x}}_{\mathit{n}},{\mathit{y}}_{\mathit{n}}\right)\right|}$

(6.15)

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

$\mathit{J}\left(\mathit{x},\mathit{y}\right)=\left|\begin{array}{ll}\frac{\partial \mathit{g}}{\partial \mathit{x}}\hfill & \frac{\partial \mathit{g}}{\partial \mathit{y}}\hfill \\ \frac{\partial \mathit{h}}{\partial \mathit{x}}\hfill & \frac{\partial \mathit{h}}{\partial \mathit{y}}\hfill \end{array}\right|=\left(\frac{\partial \mathit{g}}{\partial \mathit{x}}\right)\left(\frac{\partial \mathit{h}}{\partial \mathit{y}}\right)-\left(\frac{\partial }{}\right)$

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