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**Mysad** I am having issues with this exercise:

The life of a device is $\mathrm{Exp}(1/a)$-distributed. If one wishes to use it on $n$ different, independent, $\mathrm{Exp}(1/(na))$-distributed occasions, what is the probability that this is possible?

So I denote the life by $X$ and the time of the $n$:th occasion by $Y_n$. I have been trying to compute $P(Y_n \leq X)$ using the formula

$$ F_{Y_n|X=x}(x) = \frac{\int_0^x f_{x,y}(x,z) dz}{\int_{-\infty}^{\infty} f_{x,y}(x,z)dz} $$,

and I am no where near the answer that is $\left(\frac{n}{n+1}\right)^n.$ I have a feeling that I am way off but no idea how to get to the right answer. Or even for just $Y_1$ the answer is supposed to be $\frac{n}{n+1}$ but I just don't see how.