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If $X$ is distributed as Beta distribution and $Y$ as Chi-Squared, does the distribution of $Z = X Y$ have a name?

For instance if $X\sim \text{Beta}\left(\frac{1}{2},1\right)$ and $Y\sim \chi^2(1)$, I see the following values for pdf $f(x)$ and moment generating function $g(t)$. Curious how this relates to known distributions

$$f(z)=\frac{\Gamma \left(0,\frac{z}{2}\right)}{2 \sqrt{2 \pi } \sqrt{z}}$$

$$g(-t)=\frac{\sinh ^{-1}\left(\sqrt{2} \sqrt{t}\right)}{\sqrt{2} \sqrt{t}}\xrightarrow{t\approx\infty} \frac{\log (t)}{2 \sqrt{2} \sqrt{t}}$$

Notebook

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    $\begingroup$ In mathematics, $*$ often represents convolution, which crops up a lot in distribution questions. It's usually best to choose a different symbol for multiplication in algebraic expressions. Note that with the mgf it's behaviour near 0 that is important $\endgroup$
    – Glen_b
    Commented Jun 17, 2023 at 23:22
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    $\begingroup$ The solution is a Hypergeometric1F1 mess distribution. If you are happy to assume that $X\sim \text{Beta}(a,1)$ and $Y \sim Chi2(v)$, then the pdf solution has a reasonably tractable form as $\frac{2^{-a} a z^{a-1} \Gamma \left(\frac{v}{2}-a,\frac{z}{2}\right)}{\Gamma \left(\frac{v}{2}\right)}$ ..but still not something obviously recognisable. $\endgroup$
    – wolfies
    Commented Jun 18, 2023 at 13:20

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