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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!

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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 $$\varphi(z/y;\mu,\sigma) \beta(1-y)^{\beta-1} \times \frac{\text dx}{\text dz}= \frac{\beta}{\sqrt{2\pi\sigma^2}}\exp\{-(z-y\mu)^2/2\sigma^2y^2\}(1-y)^{\beta-1} \times \frac{1}{y}$$

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    $\begingroup$ 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? $\endgroup$
    – Dario
    Commented Jan 21, 2021 at 18:58
  • $\begingroup$ Typos corrected. $\endgroup$
    – Xi'an
    Commented Jan 21, 2021 at 19:09

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