Given a random variable $X$ being a Student's $t$ with $k$ degrees of freedom, find the distribution of $Y = X^2$.
$$f_X(x;k) = \frac{\Gamma\left(\frac{k+1}{2}\right)}{\sqrt{k\pi} \ \Gamma \left(\frac{k}{2}\right)}{\left(1+\frac{x^2}{k}\right)}^{-\frac{k+1}{2}}$$
Since the Student's $t$ doesn't have a MGF and its CF is so ugly it gives me chills I don't know what to do.
A friend of mine just showed me how to do it:
Consider $U \sim N(0,1)$ (standard normal) and $V \sim \chi^2_k$ (chi-squared distribution with $k$ degrees of freedom). The Student's $t$-distribution witg $k$ degrees of freedom is computed by $$X = \frac{U}{\sqrt{\frac{V}{k}}}$$ and $Y = X^2$ is computed by $$Y = X^2 = \frac{U^2}{\left(\sqrt{\frac{V}{k}}\right)^2} = \frac{[N(0,1)]^2}{\frac{\chi^2_k}{k}} = \frac{\chi^2_1 / 1}{\chi^2_k / k} \sim F_{(1,k)}$$ where $F_{(1,k)}$ is the Fisher-Snedecor F Distribution with $(1,k)$ degrees of freedom.
