The text I'm using on questions like these does not provide step by step instructions on how to solve these, it skipped many steps in the examples and due to such, I am rather confused as to what I'm doing when it comes to general equations like these.
Let $S, T, U$ be independent exponential random variables with common rate of 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 $Var(Y)$.
c) Find the joint distribution function of $(T,Y)$.
Any help would be appreciated.
Given the information below, you should be able to compute the answers...
i) For this you should know because it is the sum of exponential random variables is Gamma distributed (check your book for the definition of a Gamma distributed rv).
ii) For some $a > 0$, we have: $P(Y > a)$ = $P(\min(T,U)> a)$ = $P(T > a, U > a)$ = $P(T > a)P(Y > a)$. From this you can find the cdf and then the pdf.
iii) Let $Z=\max\{S,T,U\}$, then for some $t>0$ we have
$$P(Z\leq t)=P(S\leq t,T\leq t,U\leq t)=P(S\leq t)P(T\leq t)P(U\leq t)$$
Now you can find the pdf of $Z$
b) Using the results from (ii), you have the pdf. Now compute $E[Y] = \int_0^{\infty} y f_Y(y)dy$, and $Var(Y)$ = $\int_0^{\infty} y^2 f_Y(y)dy$ - $E(Y)^2$
c) Compute $P(T <t,Y < y)$