I'm sorry if this question is trivial but I'm new at this and I'm learning this stuff on my own from a book + youtube tutorials so I haven't really got anyone to ask. I have the following exercise from a book of basic probability theory:
"A process is divided into 3 sub-processes which are carried out in parallel. Each sub-process takes an Exponential amount of time, 5 minutes on the average, independent of the other sub-processes. The process is finished when all sub-processes are finished. What is the expected time for the full process to complete?"
If I understand this correctly we have three random variables $X_1, X_2, X_3 \sim Exponential(1/5)$. My naive guess would be that the expected time is 5 minutes as the sub-processes are independent and parallel but I'm probably missing something. I guess I could compute the joint distribution $f(Y) = f(X_1, X_2, X_3)$ and calculate $E(Y)$ from this. However, this entails calculating a triple integral and I'm hoping something in the simple setup (independence and identical distributions) means there is a simpler solution. Can anyone give me any advice?