# Product of a Gaussian by a Beta random variable

I'm trying to find the distribution of a random variable $$Z = X \cdot Y$$, where $$X \sim N(\mu,\sigma^2)$$ and $$Y \sim \text{Beta}(\alpha,\beta)$$ with $$\alpha$$=1.

I have tried with the convolution following this recipe https://math.stackexchange.com/a/3274225 without success. On the other hand, Meijer G-function seems to be a very general solution (https://arxiv.org/pdf/1507.07696.pdf) not easy to particularize in this simple case.

Do you have any suggestions? Thanks in advance!

The density of $$Z$$ writes directly as $$\int_0^1 \frac{\beta}{\sqrt{2\pi\sigma^2}y}\exp\{-(z-y\mu)^2/2\sigma^2y^2\}(1-y)^{\beta-1}\text dy$$ This is a consequence of the substitution rule: the pair $$(Z,Y)$$ has the joint density $$\varphi(z/y;\mu,\sigma) \beta(1-y)^{\beta-1} \times \frac{\text dx}{\text dz}= \frac{\beta}{\sqrt{2\pi\sigma^2}}\exp\{-(z-y\mu)^2/2\sigma^2y^2\}(1-y)^{\beta-1} \times \frac{1}{y}$$
• Thanks for your answer! I would expect 1 as the upper limit of the integral and $\sigma$ out of the root. Could you please give me some hint about how to reach this expression? Commented Jan 21, 2021 at 18:58