A complex machine is able to work if at least 3 of it’s 5 components work. If each motor independently functions for a random amount of time with density given by $f(x) = \frac{x}{e^x} , x>0$, compute the density function of the length of time that the machine functions.
My approach :
Let $p$ be the probability that component $i$ works for an amount of time $t$, I can find this by integrating the pdf over $0$ to $t$.
Now probability that $P(\text{at least 3 work}) = 1-P(4)^c - P(5)^c$ this I can find by assuming the above as a binomial variable.
Will this be the pdf, which is to say, is the pdf equal to $P(\text{at least 3 work})$?
I think you're close. At time $t$, the probability that a given component still works is $$p(t) = \int_0^t \tau e^{-\tau}\, d\tau = \left. -(\tau+1) e^{-\tau}\right|_{\tau=0}^{\tau=t} = 1 - (t+1) e^{-t}.$$
The chance that exactly $n$ out of 5 components still work at time $t$ is thus $$\binom{5}{n} p(t)^n (1-p(t))^{5-n}$$ and the chance that the machine works is just the sum of the above expression from $n = 3$ to $n = 5$, which is probably a bit messy but straightforward to compute.