Deduced distribution of X from X^2

Question

If I have a random variable $X^{2}$ with distribution $X^{2}∼Γ(α,β)$ then what would be the distribution of $Y=X$ ?

Thank you.

Answer 1

Our challenge is to deduce the distribution of $X$ from that of $X^2$. If $X$ is a non-negative variable, $$P(Y\le y)=P(X^2\le y^2)=\int_0^{y^2}f(t) dt,\,f(t):=\frac{\beta^\alpha}{\Gamma(\alpha)}t^{\alpha-1}\exp -\beta t.$$For $y\ge 0$< the pdf of $Y$ is then$$p(y)=\left\{ \begin{array}{ll} 0 & y<0\\ \frac{d}{dy}\int_{0}^{y^{2}}f(t)dt=2\frac{\beta^{\alpha}}{\Gamma(\alpha)}y^{2\alpha-1}\exp-\beta y^{2} & y\ge0 \end{array}\right..$$But other solutions exist. What happens if $X$ is a non-positive variable instead? Then the pdf of $Y$ is $p(-y)$. Or what if $X$ has an even distribution? Then the pdf of $Y$ is $(p(y)+p(-y))/2$. In fact, for any $q\in [0,\,1]$ we can take the pdf of $Y$ as $qp(y)+(1-q)p(-y)$.