Let $X_n$ be a sequence of iid $U[0,1]$ random variables. Let $Y_n = n \min_{1\leq i \leq n}X_i$. Show that $Y_n$ converges weakly to Exponential distribution with parameter 1.
I know that if $Z_n =\max_{1 \leq i \leq n}X_i$, then $n(1 – Z_n)$ converges weakly to Exponential distribution with parameter 1. Can I somehow use that to prove the above statement?
Best Answer
I don't think the result you know is useful here. Just note that $P(Y_n >x)=P(X_1 >\frac x n)^{n}=(1-\frac x n)^{n} \to e^{-x}$ as $ n \to \infty$ for all $x>0$. Hence $Y_n$ converges to exp(1) in distribution.
The first equality comes from the fact the minimum of $n$ numbers is greater than $x$ iff each of the numbers is greater than $x$, followed by the fact that $X_i$' s are i.i.d..