= Exponential endpoint limit of a uniform maximum
{title2=$n(1-M_n)\xrightarrow{\mathrm d}\operatorname{Exp}(1)$}
For <independent and identically distributed random variables> $U_i$ with the <continuous uniform distribution> on $[0,1]$, let $M_n=\max_{1\leq i\leq n}U_i$. Its <cumulative distribution function> is $x^n$ on $[0,1]$, so $M_n\xrightarrow{\mathrm d}1$. The endpoint error has a nonconstant limiting <probability distribution>: for fixed $z\geq0$ and $n\geq z$,
$$
\mathbb P(n(1-M_n)>z)=(1-z/n)^n\longrightarrow e^{-z}.
$$
This <survival function> gives the displayed <exponential distribution> limit. More generally, for observations uniform on $[a,b]$, $n(b-M_n)/(b-a)$ has the same limit. The scaling identifies an error of order $1/n$, in contrast to the $1/\sqrt n$ fluctuations of a typical <sample mean> with nonzero finite <variance>.
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