Proof: By Property 1 of Wilcoxon Rank Sum Test, R1 + R2 = n(n+1)/2. Thus
Property 2: For n1 and n2 large enough the U statistic is approximately normal N(μ, σ) where
Proof: From Property 2 of Wilcoxon Rank Sum Test, the mean of R1 is . Similarly the mean of R2 is . Since
it follows from Property 3 of Expectation that the mean of U1 is
Similarly the mean of U2 is . Since U = min(U1, U2), it follows that the mean of U is also .
Similarly the variance of U2 is the same as the variance of R2, which is again . Thus the variance of U is this same amount.