Product of two exponentially distributed random variables

Question

I am trying to find the close form expression of probability distribution of $Z$ such as $Z=X_1X_2$ where $X_1$ and $X_2$ are two independent exponentially distributed variables with PDF

$P(x1)=λ_1e^{−λ_1x_1}$; $P(x2)=λ_2e^{−λ_2x_2}$

I know the PDF of $Z$ is

$λ_1 λ_2 \int_0^\infty \exp({-λ_1x_1-λ_2z/x_1)}dx_1$

My question is, is there any easy way to solve the integration part or do we have any close form? Please help.

Answer 1

The PDF of $X_1\cdot X_2$ depends on the Bessel K function, since: $$\int_{0}^{+\infty}\exp\left(Ax-Bz/x\right)\,dx = 2\sqrt{\frac{Bz}{A}}\cdot K_1\left(2\sqrt{ABz}\right).$$

Answer 2

Closed form integration of above integral can be found in $Eq. 3.324.1$

Given as \begin{equation} \int_{0}^{\infty}\exp\bigg(-\frac{\beta}{4x}-\gamma x\bigg)dx= \sqrt{\frac{\beta}{\gamma}}K_1\big(\sqrt{\beta \gamma}\big) \end{equation} from the book by I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, 8e