Find the limiting distribution of $nY_n$ where $Y_n = \min(X_1, ..., X_n)$ and $X_1, ..., X_n\sim \operatorname{unif}(0,2)$ are uniformly distributed random variables.
Here is what I did:
$$P(nY_n > x) = P(Y_n > x/n) = (P(X_1 > x/n))^n = \left(\int_{x/n}^2 \frac{1}{2} dx\right)^n = \left(1 - \frac{y}{2n}\right)^n$$
As $n \to \infty$, this probability converges to $e^{-y/2}$
And the distribution would be $$ 1- e^{-y/2} $$ which is $\operatorname{Exp}(1/2)$.
Let me know if I made any mistakes, I am simply checking if I did this correctly.