Q: Ratio between normal distributed and gamma distributed variables Let $X \sim N(0,1)$ and $G \sim Gamma(a)$. Why is $\frac{X}{G}$ t-distributed? 

A: Gamma(a) is nothing but a chi-square distribution. And you know t-distribution is ratio of Standard normal and chi-square distribution upon number of degree of freedom.
From this relation you will get answer.
