Finding joint distribution function of dependent random variables

Question

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 $(T,Y)$.

Answer 1

$$ F_{T,Y}(t,y)=\mathbb P(T\leq t,\ Y\leq y)=\mathbb P(T\leq t,\ \min\{T,U\}\leq y). $$ For $t\leq y$ first event implies the second one, and you can find this probability directly.

Let us consider $t>y$ and use the property $\mathbb P\bigl(AB\bigr)=\mathbb P\bigl(A\bigr)-\mathbb P\bigl(A\overline B\bigr)$: $$ \begin{align} F_{T,Y}(t,y) &=\mathbb P(T\leq t,\ \min\{T,U\}\leq y) \cr &=\mathbb P(T\leq t)-\mathbb P\bigl(T\leq t,\ \min\{T,U\}> y\bigr)\cr & =F_T(t)-\mathbb P\bigl(T\leq t,\ T>y, U> y\bigr). \end{align} $$ Next use independence of $T$ and $U$ and find the last probabililty.