TITLE: Find joint distribution of minimum and maximum of iid random variables
QUESTION [6 upvotes]: $(X_n)$ sequence of iid random variables with uniform distribution $U([0,1])$.
$m=\min(X_1,...X_n), M=\max(X_1,...X_n)$.
I want to find $f_{m,M}(s,t)$.
$$
\begin{split}
P(m<s,M<t)
 &= P(m<s)P(M<t)1_{m\ne M}+P(X_1<\min(s,t))1_{m=M} \\
 &= (1-(1-s))^nt^n1_{m \ne M}+\min(s,t)1_{m=M} \\
 &=((st)^n+s)1_{s<t}+((st)^n+t)1_{s \ge t}
\end{split}
$$
When I differentiate it, I get $f_{m,M}(s,t)=n^2t^{n-1}s^{n-1}$.
Is this okay? And does it mean that $M$ and $m$ are independent and $f_{m,M}(s,t)=f_m(s)f_M(t)$?

REPLY [15 votes]: For a sequence of $n$ iid continuous samples, $(X_i)_{n=1}^n$, the minimum is less than $s$ and maximum less than $t$ iff all samples are less than $t$ and at least one is less than $s$.
$$\begin{split}
\mathsf P(m\leqslant s, M\leqslant t) &=\mathsf P\Big(\big(\bigcup_{i=1}^n \{X_i\leqslant s\}\big)\cap\big(\bigcap_{i=1}^n\{X_i\leqslant t\}\big)\Big)
\\&= \mathsf P\Big(\big(\bigcap_{i=1}^n\{X_i\leqslant t\}\big)\setminus\big(\bigcap_{i=1}^n\{s<X_i\}\big)\Big)
\\ &= \mathsf P\big(\bigcap_{i=1}^n\{X_i\leqslant t\}\big)-\mathsf P\big(\bigcap_{i=1}^n\{s< X_i\leq t\}\big)
\\ &= \prod_{i=1}^n\mathsf P\{X_i\leqslant t\}-\prod_{i=1}^n\mathsf P\{s<X_i\leqslant t\}
\\ &= \big(\mathsf P\{X_i\leqslant t\}\big)^n-\big(\mathsf P\{s<X_i\leqslant t\}\big)^n
\\ & =\begin{cases} 0 &:& s<0 ~\vee~ t<0
\\ t^n-(t-s)^n & :& 0\leqslant s\leqslant t< 1 
\\ t^n &:& 0\leqslant t < \min (s,1)
\\ 1-(1-s)^n &:& 0\leqslant s< 1\leqslant  t
\\ 1 &:& 1\leqslant s ~\wedge~ 1\leqslant t 
\end{cases}
\\[2ex]f_{n,M}(s,t) &=\begin{cases} n(n-1)(t-s)^{n-2} & :& 0\leqslant s\leqslant t< 1 
\\ 0 &:& \textsf{elsewhere} 
\end{cases} 
\end{split}$$