# Expectation of a parallel system

A system consists of $n$ components in parallel. The lifetimes of the components are i.i.d. exp($\lambda$) random variables. The system functions as long as at least one of the $n$ components is functioning. Let $T$ be the lifetime of the system. Compute $E[T]$.

Ok, so I've done problems like these in the past and they usually involved a finite number of components with a fixed probability. That is, the components usually had a fixed probability $p$ of failure. So I would just plug that into a Binomial distribution and call it a day. This problem is a bit different because $p$ varies with time.

So I do have some ideas for this problem, but I eventually get stuck.

Because this is a Poission process, the PP is counting the number of failures up to time $T$. So because the system fails when the last component fails, we essentially want to find the expected lifetime of the last component. In math terms:

Let $N(t)$ = the number of failures by time $t$
Let $T$ = the lifetime of the system
Let $S_i$ = the lifetime of component $i$