Fundamentals of Applied Probability and Random Processes, 2nd Edition

6.8 Two Functions of Two Random Variables

Let X and Y be two random variables with a given joint PDF f_XY(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, f_UW(u, w), in terms of the PDFs of X and Y.

It can be shown that if (x₁, y₁), (x₂, y₂), …, (x_n, y_n) are the real solutions to the equations u = g(x, y) and w = h(x, y) then f_UW(u, w) 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(x, y) is called the Jacobian of the transformation {u = g(x, y), w = h(x, y)} and is defined by

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