The life of a repairing device is Exp(1/a)-distributed. Peter wishes to use it on n different, independent, Exp(1/na)-distributed occasions.
(a) Compute the probability Pn that this is possible.
(b) Determine the limit of Pn as $n \to \infty$
. Part (b) is easy if you have (a), which should be $( \frac{n}{n+1})^n$
My first source of confusion is $Exp(\frac{1}{an})$. So this should be $(an)e^{-anx}$ ? I'm guessing that we need to "condition" on this and I get something like:
$$ \int_0^\infty ( a e^{-ax})(an e^{-anx} ) dx = \frac{a^2 n}{a+an} = \frac{an}{1+n}$$
which at least looks somewhat like the answer, though it completely wrong. I have no clue how the occasions can be continuously Exp distributed if "n" should be a discrete quantity.