The PDF of a Function of Two Random Variables

Shoichi Midorikawa

General Formula

　Let $X$ and $Y$ be two independent continuous random variables. Suppose $Z$ is a function of $X$ and $Y$ , i.e., $z = \varphi(x, y)$ , then the pdf of $z$ is $f_Z(z) = \int \delta \left(z-\varphi(x, y) \right) f_X(x) f_Y(y) \, dx \, dy$ where $\delta \left(z-\varphi(x, y) \right)$ is the Dirac delta function.

　To evaluate the integral, it is convenient to use the formula $\delta\left( g(x) \right) = \sum_i \frac{1}{|g'(\alpha_i)|}\delta(x-\alpha_i)$ where $\alpha_i$ is a solution of $g(x)=0$ , namely, $g(\alpha_i)=0$ .

　The following are representative examples.

1. $\bf Z=X+Y$

$\begin{eqnarray*} f_Z(z) &=& \int \delta(z-x-y)f_X(x)f_Y(y)\, dx\, dy \\ &=& \int f_X(z-y) f_Y(y)\, dy \end{eqnarray*}$

2. $\bf Z=X-Y$

$\begin{eqnarray*} f_Z(z) &=& \int \delta(z-x+y)f_X(x)f_Y(y)\, dx\, dy \\ &=& \int f_X(z+y) f_Y(y)\, dy \end{eqnarray*}$

3. $\bf Z=XY$

$f_Z(z) = \int \delta(z-xy)f_X(x)f_Y(y)\, dx\, dy \\$ Note that

$\delta(z-xy) = \delta\left(y(x-\frac{z}{y})\right)= \frac{1}{|y|}\delta\left(x-\frac{z}{y}\right)$ Thus we find

$f_Z(z) = \int \frac{1}{|y|}f_X(z/y) f_Y(y)\, dy$

4. $\bf Z=X/Y$

$f_Z(z) = \int \delta\left(z-\frac{x}{y}\right)f_X(x)f_Y(y)\, dx\, dy \\$ Note that

$\delta\left(z-\frac{x}{y}\right) = \delta\left(\frac{1}{y}(x-yz)\right)= |y|\delta\left(x-yz\right)$ We thus find

$f_Z(z) = \int |y| f_X(yz) f_Y(y)\, dy$

5. $\bf Z=X^Y$

$f_Z(z) = \int \delta(z-x^y)f_X(x)f_Y(y)\, dx\, dy \\$

${\rm Let} \ \ g(x) = z-x^y.$

${\rm Then, \ \ from}\ \ g(x)= 0, \ \ x = z^{1/y} .$

${\rm From}\ \ g'(x) = -yx^{y-1},$

$g'(z^{1/y}) = - y z^{1-1/y}.$ We finally find

$f_Z(z)= \int \frac{1}{|yz^{1-1/y}|}f_X\left(z^{1/y}\right) f_Y(y)\, dy$

For more details, see PDFs of X + Y, XY, X/Y , and XY