# The distribution of minmax and maxmin deviations of a Random variable

Let $X_1,X_2,X_3,......,X_n$ be $n$ independently and uniformly distributed random variables in the interval $[a,b]$.

Further let $P=\min \{X_i,i=1,2,3..,n\}$ and $Q=\max\{X_i,i=1,2,3..,n\}$. Someone please help me in finding the distribution of the following random variables:

$$U=\max|X_i-P| \text{ and } V=\min|X_i-Q|,$$ with $i$ ranging from $1$ to $n$.

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As $P$ is the sample min, $U$ is just the sample range. And as Q is one of the $X_i$, and you seek to minimise the deviation around $Q$, it follows that $V = 0$. –  wolfies Jun 6 '14 at 14:25