Chapter 6, Problem 6.33TE

Let $X_1,..\mathrm{..}{X}_n,$ be independent and identically distributed random variables having distribution function F and density f. The quantity $M\equiv \frac{\left[X_{\left(1\right)}+X_{\left(n\right)}\right]}{2}$ defined to be the average of the smallest and largest values in $X_1,..\mathrm{..}{X}_n,$ is called the midrange of the sequence. Show that its distribution function is $F_M\left(m\right)=n{\displaystyle {\int }_{-\infty }^m{\left[F\left(2m-x\right)-F\left(x\right)\right]}^{n-1}f\left(x\right)dx}$