Say we have the sample $X_{(1)} < X_{(2)} < X_{(3)}$ from the distribution $f_X(x) = 2x$, $0<x<1$.
What is the probability that the minimum and the maximum are between $0.1$ and $0.7$?
My approach,
Let $X = X_{(1)}$ and $Y = Y_{(3)}$. Then $f_{X, Y}(x, y) = 48(xy^5 - 2y^3x^3 + x^5y)$
We want $$\int_{0.1}^{0.7}\int_{0.1}^{0.7}48(xy^5 - 2y^3x^3 + x^5y)dxdy$$
However, this integration becomes very messy. Is there a better alternative?
Hint: Do not focus on the distribution of the order statistics. Look to the original samples.
The probability that the minimum and maximum of three independent and identically distributed random variables are between two numbers, is the probability that all of them are between those numbers.