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Let $X_1,\ldots, X_n \sim U[0, 1]$. Let $Y_n = \min_{1 \leq i\leq n} X_i$. Show that $nY_n$ converges in distribution to some random variable $W$ . Find the distribution of $W$ explicitly.

I know that $$F(W)=P(W\le w)=P(nYn\le w)=P(Yn\le w/n)=1-(1-w/n)^n$$

But I cannot continue to finish the question.

StubbornAtom
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2 Answers2

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Slightly less elementary than the other answer, but I believe that these facts are also useful to know. I'll give the outline of the proof, necessary steps aren't that hard to do:

1) If $\{ X_i \} \sim U(0, 1)$ and iid, then $\min X_i \sim B(1, n)$, which is Beta random variable.

2) It can be shown using convergence of moments that $n B(1,n) \rightarrow G(1, 1)$, where $G(\alpha, \beta)$ is Gamma random variable.

defenestrator
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Hint: Use the fact that $\lim\limits_{n\to\infty}\left(1 +\frac{x}{n}\right)^n = e^x$ for any $x\in\mathbb{R}$.

Minus One-Twelfth
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