TITLE: PDF of Ratio of Normal and Gamma Random Variables
QUESTION [1 upvotes]: Let $X \sim N(0,1)$ and $Y \sim \Gamma\left(\frac{k}{2}, \frac{1}{2}\right)$. If $X$ and $Y$ are independent, find the pdf of $$V=\frac{X}{\sqrt{Y/k}}.$$
For this problem, I introduced $U=X$, and attempted to find the joint pdf of $U$ and $V$. However, as I was getting the marginal pdf of $V$, I got stuck. Can anyone help me on this?

REPLY [0 votes]: $\Gamma\left(\frac{k}{2},\frac{1}{2}\right)$ is the same distribution as $\chi^2_k$. Hence the given statistic has Student's $t$ distribution with $k$ degrees of freedom. See t-distribution wiki
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