TITLE: Product of two random variables that has exponential distribution
QUESTION [9 upvotes]: I am trying to define the probability distribution of $Z$ such as $Z = X_1\cdot X_2$ where $X_1$ and $X_2$ are two independent and identically exponentially distributed variables.
$$P(X_1=x) = \lambda e^{-\lambda x}$$
$$P(X_2=x) = \lambda e^{-\lambda x}$$
I tried something like that….
$$P(Z=x) = P(X_1 = a) \cdot P(X_2 = x/a)$$
Is this first equation correct?
$$P(Z=x) = \lambda e^{-\lambda a} \cdot \lambda e^{-\lambda x/a} = \lambda^2 e^{-\lambda (a+x/a)}$$
I doesn't feel like I found the solution as the dummy variable $a$ remains in the final result. Can you please help me solving this problem?

REPLY [7 votes]: The density of $X_1 X_2$, obtained from that integral others have posted, is $f(z) = 2 \lambda^2 K_0(2 \lambda \sqrt{z})$ for $z > 0$, $0$ otherwise, where $K_0$ is a modified Bessel function of the second kind.