I'm not sure how to do part (c) of this question. I can't just multiply the distribution functions of T and Y because they are dependent right?
Let $S$, $T$, $U$ be independent exponential random variables with common rate $2$.
(a) Find the probability density functions for
i. $X = S+T+U$;
ii. $Y = \min\{T,U\}$;
iii. $Z = \max\{S,T,U\}$.
(b) Compute $E[Y]$ and $\operatorname{Var}(Y)$.
(c) Find the joint distribution function of $(T,Y)$.