Having $X_i$ i.i.d. uniform r.v. in $[0, \theta]$ for $\theta > 0$ and $M_n = \max(X_i)$,
I would like to compute the cumulative distribution function $F_n(t)$ of $n(1−M_n/θ)$ for fixed $t∈[0,n]$ and any positive integer $n$.
In my understanding $F_n(t) = P(n \leq t) = (t/\theta)^n$.
From there, how do I continue? Plugging in $n(1−M_n/θ)$ for $t$ does not seem the right approach with $(n(1/\theta-M_n))^n$ as answer.
With the obtained value, I would need to compute the limit $$\lim_{n\to\infty} F_n(t)$$
Thanks for guiding me here.
The limit result is an $Exp(1)$ rv
In fact,
$Y_i=\frac{X_i}{n}\sim U(0;1)$
So letting $Y_n=\frac{M_{n}}{\theta}$
$F_{\frac{M_{n}}{\theta}}(t)=\mathbb{P}[n(1-Y_{n})\leq t]=$
$=\mathbb{P}[Y_{n}>1-\frac{t}{n}]=1-(1-\frac{t}{n})^n=(1-e^{-t})\mathbb{1}_{(0;\infty)}(t)$
When $n \rightarrow \infty$