Let X1, . . . , Xm and Y1, . . . , Yn be the observations

Question:

Let X1, . . . , Xm and Y1, . . . , Yn be the observations in two samples, and suppose that no two of these observations are equal. Consider the mn pairs
(X1, Y1) . . . (X1, Yn),
(X2, Y1) . . . (X2, Yn),
.
.
.
(Xm, Y1) . . . (Xm, Yn).
Let U denote the number of these pairs for which the value of the X component is greater than the value of the Y component. Show that
U = S – 1/2 m(m + 1),
where S is the sum of the ranks assigned to X1, . . . , Xm, as defined in this section.