Quick Question Integration with Joint PDF

Question

Let $X_1, X_2, \ldots, X_n$ by independent and identically distributed random variables with probability density function (pdf) $$f_X(x) = \left\{\begin{array}{ll}1, & 0 < x < 1\\ 0, & \text{otherwise.}\end{array}\right.$$

1. Show that the joint pdf of $X_{(1)} = \min\{X_1, X_2, \ldots, X_n\}$ and $X_{(n)} = \max\{X_1, X_2, \ldots, X_n\}$ is $$f_{X_{(1)},X_{(n)}}(x, y) = \left\{\begin{array}{ll}n(n-1)(y-x)^{n-2}, & 0 < x < y < 1\\ 0, & \text{otherwise}\end{array}\right.$$
2. Let $R = X_{(n)} - X_{(1)}$ be the range of the random variables $X_1, X_2, \ldots, X_n$.
   By using the formula $$ P(R \le u) = \iint\limits_{y \le x + u} f_{X_{(1)},X_{(n)}}(x, y)\, dy\, dx, $$ show that $P(R \le u) = n(1 - u)u^{n-1} + u^n$, $0 < u < 1$.

For part 2, what are the upper and lower limits of $Y$ and $X$?

I tried $x$ from $0$ to $u$ and $y$ from $0$ to $x + u$. And this doesn't work.

Please help. Could you please help me out if you see this please ?

Answer 1

For convenience, let $g(x, y) = f_{X_{(1)}, X_{(n)}}(x, y)$. The region is:

\begin{align} P(R \le u) & = \iint\limits_{y \le x + u}g(x, y)\, dy\, dx\\ & = \int\limits_0^{1-u} \int\limits_x^{x+u} n(n−1)(y−x)^{n−2}\, dy\, dx + \int\limits_{1-u}^1 \int\limits_x^1 n(n−1)(y−x)^{n−2}\, dy\, dx\\ & = n(1 - u)u^{n - 1} + u^n,\ 0 < u < 1 \end{align}