= Uniformized sample range
{title2=$F(X_{(n)})-F(X_{(1)})\sim\operatorname{Beta}(n-1,2)$}
For an <independent and identically distributed>[iid] sample of size $n\geq2$ from a continuous <cumulative distribution function>, the <probability integral transform> makes the transformed sample uniform. Integrating the <joint probability density>[joint density] of its minimum and maximum gives range <probability density function>[density] $n(n-1)r^{n-2}(1-r)$ on $0<r<1$. Its <cumulative distribution function>[distribution function] is $nr^{n-1}-(n-1)r^n$. The result describes population content between the extreme <order statistics>, <independence>[independently] of the underlying continuous <probability distribution>[distribution].
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