= Maximum of independent exponential random variables
{title2=$M_n=\max_{1\leq i\leq n}X_i,\quad X_i\sim\operatorname{Exp}(\lambda)$}
= Maxima of independent exponential random variables
{synonym}
Let $n\geq1$ be an <integer> and let $X_1,\ldots,X_n$ be <independent and identically distributed random variables> with an <exponential distribution> of rate $\lambda>0$. Their <sample maximum> $M_n$ is nonnegative. Substituting the <exponential distribution>'s <cumulative distribution function> into the <distribution functions of independent minima and maxima> gives the exact law
$$
\mathbb P(M_n\leq x)=
\begin{cases}
(1-e^{-\lambda x})^n,&x\geq0,\\
0,&x<0.
\end{cases}
$$
For example, at $x=(\log2)/\lambda$ each observation is below the threshold with <probability> $1/2$, but all $n$ observations are below it with <probability> $2^{-n}$. Thus an individual observation's <median> becomes an increasingly unlikely upper bound for the whole batch. The general <order statistic> density in Siegrist's treatment, with $k=n$, also gives this law on <integration>. https://www.randomservices.org/random/sample/OrderStatistics.html[Siegrist, Random Services: exponential order statistics]{ref}.
Solving $(1-e^{-\lambda x})^n=p$ gives the exact <quantile function>, for $0<p<1$,
$$
q_n(p)=-\frac1\lambda\log(1-p^{1/n}),\qquad
m_n=q_n(1/2)=-\frac1\lambda\log(1-2^{-1/n}).
$$
In a toy model with $\lambda=1\,\mathrm{s}^{-1}$, the <median> maximum is $0.693147$ seconds for $n=1$, $2.703555$ seconds for $n=10$, and $4.975147$ seconds for $n=100$. All three use the exact <distribution function>; the increase comes from taking a <maximum> over more observations. The $n=2$ case connects to the existing <maximum of two exponentials as a scaled sum>.
\Image[/exponential-sample-maximum-distributions.png]
{title=Exact cumulative distribution functions of exponential sample maxima, with rate one per second and median markers}
The <Gumbel distribution> explains the logarithmic shift visible here: after subtracting $(\log n)/\lambda$ and measuring deviations in units of $1/\lambda$, the <distribution functions> approach one fixed shape. The next derivation makes that <convergence in distribution> explicit.
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