Expectation and variance of Wilcoxon statistic. If the X’s and Y ’s are samples from continuous distributions F and G, respectively, the expectation and variance of the Wilcoxon statistic U defined in the preceding problem are given by E

 U mn

= P{X < Y } = 

F dG (6.62)

and mnV ar  U mn

=



F dG + (n − 1)



(1 − G)

2 d F (6.63)

+(m − 1)



F2 dG − (m + n − 1)



F dG 2

.

Under the hypothesis G = F, these reduce to E

 U mn

= 1 2

, V ar  U mn

= m + n + 1 12mn . (6.64)