Distribution of the Ratio of Two Independent Exponentials

Let $X$ and $Y$ be independent $\operatorname{Exp}(1)$ random variables. Define $R = X/Y$.

(a) Using the transformation $(R, S) = (X/Y,\, Y)$, compute the joint density $f_{R,S}$ via the Jacobian and then marginalize over $S$ to find the PDF of $R$.

(b) Identify $f_R$ as a named distribution and verify by computing $P(R \le 1)$ using a symmetry argument.