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Suppose that X (X1,X2, . . . , X.) ald Y-(Yİ,%, , Y,n) are random samples from continuous distributions F and G, respectively. Wilcoxons two-sample test statistioc W- W(X,Y) is defined to be X/Ri where Ri is the rank of Y in the combined sample. 1. Let T Z, where Z, Z2,, Zm are numbers sampled at random without replacement from the set {1,2,...,N) Show that E(Z) = (N + 1)/2 and hence E(T) = m(N + 1)/2 Show that 2N2 +3N +1 and E(Z,Z) 3N2 +5N +2 E(Z) = 12___ , iメj, and hence rn rn m(N-m)(N + 1) Var(T) Var(Z) + Cov(Z, Z) 12 İFJ Deduce that under the null hypothesis that FG, the expectation and variance of Wilcoxons two-sample test statistic are m(n+m+1)/2 and nm(n+m+1)/12, respectively
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NON-D 2. m(N+ 12N 12 Cov (Z N41 1 2. i 2 2 12 replacemen /2

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