Suppose the random variable e has cdf F(t). Let (u) =12[u (1/2)],0 < u < 1, denote the Wilcoxon score function. (a) Show that

Suppose the random variable e has cdf F(t). Let ϕ(u) =√12[u − (1/2)],0 < u < 1, denote the Wilcoxon score function.
(a) Show that the random variable ϕ[F(e_i)] has mean 0 and variance 1.
(b) Investigate the mean and variance of ϕ[F(e_i)] for any score function ϕ(u) which satisfies ∫¹₀ ϕ(u) du = 0 and ∫¹₀ ϕ²(u) du = 1.