Chapter 5.5, Problem 12E

Let the independent random variables $X_1$ and $X_2$ be $N\left(0,1\right)$ and $x^2\left(r\right)$, respectively. Let $Y_1=\frac{X_1}{\sqrt{\frac{X_2}{r}}}$ and $Y_2=X_2$.

(a) Find the joint pdf of $Y_1$ and $Y_2$.

(b) Determine the marginal pdf of $Y_1$ and show that $Y_1$ has at distribution. (This is another, equivalent, way of finding the pdf of T.)