Product of two independent continuous random variables

**basovic88** · November 18th 2011, 04:49 AM

I have two continuous random variables X and Y. X follows an Exponential distribution with rate $\lambda$ , and Y follows a Normal distribution with $\mu$ and $\sigma^2$ . The domain of X is $[0,\infty)$ , and that of Y is $(-\infty,\infty)$ .

I have come up with the following expression for the pdf of V=XY.
Using the result of Rohatgi (1976, p. 141)

$f_V(v)=\frac{\lambda}{\sigma_\alpha\sqrt{2\pi}} \int_0^\infty e^{-\left(\lambda x+\frac{(\frac{v}{x}-\mu_\alpha)^2}{2\sigma_\alpha^2}\right)}\frac{1}{x }\,dx$

Do you have any suggestions on how to solve this (analytically)?

**basovic88** · November 18th 2011, 12:52 PM

Added my Rohatgi result to the post.

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