Expectations of Max Order Statistic on Exponential distribution

Question

Let $X_1,...,X_n$ be iid exponential with scale parameter $\theta$. That is for each $i \in \{ 1,2,...,n\}$, $X_i \sim f_{X_i}(x) = \frac{1}{\theta}\exp(-x/\theta) $. I am interested in the max order statistic $X_{(n)} := \max_{i\in \{1,...,n\}}X_i $. I know the pdf of this max order statistic is $$f_{X_{(n)}}(x) = \frac{n}{\theta} \exp(-x/\theta)(1-\exp(-x/\theta))^{n-1} .$$

For this statistic, I wish to compute the expected value and variance. However, the actual integrals are complicated, and I cannot see how to relate this pdf to an already known distribution.

Can anyone help me figure this out?

Thank you.

Not sure it eases the computation, but if you sample the $X_i's$ as $-\theta\log(U_i)$ with $U_i$ i.i.d. Uniform$(0,1)$ r.v., then $X_{(n)}=-\theta\log\min_{i=1}^nU_i$. Thus $X_{(n)}$ has the same distribution as $-\theta\log B_n$ where $B_n$ is a Beta$(1,n)$ r.v. So $\Bbb E[X_{(n)}]=-\theta,\Bbb E[\log B_n]$, $\mathbb E[X_{(n)}^2]=\theta^2,\Bbb E[\log^2 B_n]$, etc. — nejimban, Jan 24 '22 at 16:07
I think the i'th order statistic from from exponential variables is again an exponential variable. — Snildt, Jan 24 '22 at 16:34
Check out https://math.stackexchange.com/q/3560742/321264 and its linked posts. — StubbornAtom, Jan 24 '22 at 17:13
Does this answer your question? Variance of the maximum of n independent exponentials — StubbornAtom, Jan 26 '22 at 10:16

Expectations of Max Order Statistic on Exponential distribution

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