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Given a Beta Prime distributed random variable $X \sim BP(a,b) $ with probability density $$\rho_X(x) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\frac{x^{\alpha - 1}}{(1+x)^{a+b}}, x > 0$$ Consider the distribution of $$Y = K \sqrt{X}$$ where $K > 0$ is a constant.

Does the PDF of this distribution have a closed form? If so, what does it look like?

Pame
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  • Could you explain what you mean by a "closed form"? Of what--the PDF, CDF, CF, CGF, MGF, something else? If you accept that the distribution of $X$ has a closed form for the PDF or CDF, then perforce that of $Y$ does also: just do the substitution. Conversely, if that of $Y$ has a closed form, the inverse substitution gives a closed form for $X.$ – whuber Sep 14 '22 at 16:28
  • @whuber I edited my post now to make it more clear. What I'm looking for is just the what the PDF of $Y$ looks like and if it has a name. – Pame Sep 14 '22 at 16:46
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    Your question is answered in many threads, such as https://stats.stackexchange.com/questions/14483. Depending on the values of $a$ and $b$ it has many possible names; for instance, when $a=1/2$ its a scaled Student t distribution. It is a particular instance of a Generalized Beta of the Second Kind. It's a particular instance of a Pearson distribution. Etc. – whuber Sep 14 '22 at 16:48

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