Density function of a product of random variables

Question

If $X\geq 0$ and $Y\geq 0$ are independent continuous random variables with density functions $f$ and $g$, find the density function of $XY$. I've found that the distribution function of $XY$ is \begin{align*} \mathbb {P}(XY\leq z)=\iint 1_{(xy\leq z)} \, dF(x)\,dG(y)=\int F\left(\frac{z}{y}\right)\, dG(y) \end{align*} for $F$ and $G$ be the distribution functions of $X$ and $Y$ respectively. I wonder how to differentiate $\mathbb {P}(XY\leq z)$ with respect to $z$ to find its density. And what is the density if $X$ and $Y$ are independent exponentially distributed random variables with parameter $\lambda$?

Answer 1

If $Z:=X \cdot Y$, and $X$ and $Y$ independent, then $Z$ is called a "Product Distribution"

$f_Z(z) = \frac{dF_Z(z)}{dz} = \frac{d}{dz}\int_{y=0}^{\infty} g(y) F\left(\frac{z}{y}\right)dy = \int_{y=0}^{\infty} g(y) \frac{d}{dz}F\left(\frac{z}{y}\right)dy = \int_{y=0}^{\infty} g(y) f\left(\frac{z}{y}\right)\frac{1}{y}dy $.

Using the fact that $f(x) = \lambda e^{-\lambda x}$ and $g(y) = \lambda e^{-\lambda y}$, you can easily find the product density.