Let X1,... ,Xn be independent and identically distributed random variables having distribu...

Let X1,... ,Xn be independent and identically distributed random variables having distribution function F and density f. The quantity M = [X(1)+ X(n)]/2, defined to be the average of the smallest and largest values in X1,...,Xn, is called the midrange of the sequence. Show that its distribution function is