What's the PDF of product of two Exponential Random variables

Question

I have two random variable X and Y. They follow exponential distribution with parameter $\lambda_1$ and $\lambda_2$, i.e., $X\sim λ_1e^{−λ_1x}$ and $Y\sim λ_2e^{−λ_2y}$. I wish to find the PDF of $Z=X(aY+b)$ given $a>0$ and $b>0$. Without $a$ and $b$, I can calculate using mathematica that $Z=2λ_1λ_2K_0(2\sqrt {λ_1λ_2z})$. But now I find it difficult to integrate it with $a$ and $b$. I get stuck here. Any help would be much appreciated. Thank you guys!

Answer 1

Hint 1: if $X \sim \exp(\lambda)$, then $k\cdot X\sim \exp(\frac{\lambda}{k} )$, for $k>0$.

Hint 2: if $X$ and $Y$ are independent, then their product distribution $Z=XY$ has density $f_Z$ as follows: $ f_Z(z)=\int_{-\infty}^\infty f_X(x)\cdot f_Y(x/z) \frac{1}{|x|} dx. $

See https://en.wikipedia.org/wiki/Product_distribution for more details.

As you can see, some work is required.