Say, we have a sequence of random variables $(X_i)_{1\leq i \leq n}$ that are i.i.d. and distributed according to a uniform distribution defined on interval $[0,1]$.
Why is $n-n\cdot\max\{X_1,\cdots,X_n\}$ distributed like an exponential distribution for real and positive $n$?
I know that $\lim_{n\rightarrow\infty}e(x)=(1+\frac{x}{n})^n$…
I also know that $\max\{X_1,\cdots,X_n\}=P(X_1)\cdot\cdots\cdot P(X_n)$ for independent variables…
I have:
\begin{align}
F(x) & =P(n-n\cdot\max\{X_1,\cdots,X_n\}<x)=P(\max\{\ldots\} \\[10pt]
& \geq1-x/n)=1-P(\max\{\ldots\}<1-x/n).
\end{align}
Now, due to the cdf of $U(0,1)$ which is just $x$, this is equivalent to $1-(1-x/n)^n$.
But how do I proceed now? Or are there any mistakes so far?
Best Answer
Let $M_n = \max_{1\leqslant i\leqslant n}X_i$ for each positive integer $n$. For $0<t<1$ we have $$\{M_n \leqslant t\} = \bigcap_{i=1}^n \{X_i\leqslant t\},$$ so by independence it follows that $$ \mathbb P(M_n\leqslant t) = t^n. $$ Let $Y_n = n(1-M_n)$. Let $t>0$ and choose $N>1/t$. Then for $n\geqslant N$ we have \begin{align} \mathbb P(Y_n\leqslant t) &= \mathbb P(n(1-M_n)\leqslant t)\\ &=\mathbb P(1-M_n\leqslant t/n)\\ &=\mathbb P(M_n\geqslant 1-t/n)\\ &=1-\mathbb P(M_n\leqslant 1-t/n)\\ &=1 - (1-t/n)^n. \end{align} Since $$\lim_{n\to\infty}1 - (1-t/n)^n = 1 - e^{-t},$$ it follows that $Y_n$ converges in distribution to the exponential distribution with mean $1$.