$X_1, \ldots, X_n$ are independent random variables from a continuous distribution function $F$. I would like to calculate the probability that $P(((X_{(1)},X_{(n)})\subset(q_{1/4},q_{3/4}))$. Intuitively I would say it is $(\frac{1}{2})^n$. But I want some how to "proof" that using
$P(X_{(k)}\leq x) = 1 - F_{n,F(x)}(k-1)$. But I cant handle the open boundries... What I have so far:
$P((X_{(1)},X_{(n)})\subset(q_{1/4},q_{3/4})) = 1 -P((X_{(1)},X_{(n)})\not\subset(q_{1/4},q_{3/4})) $