Suppose X is a random variable with mean 0 and variance 2 . Recall that the function F x, (t) is the cdf of

Suppose X is a random variable with mean 0 and variance σ². Recall that the function F_x,ϵ(t) is the cdf of the random variable U = I_1−ϵX + [1 − I_1−ϵ]W, where X, _1−ϵ, and W are independent random variables, X has cdf F_X(t), W has cdf Δ_x(t), and I_1−ϵ has a binomial(1,1 − ϵ ) distribution. Define the functional Var(F_X) = Var(X) = σ². To derive the influence function of the variance, perform the following steps:
(a) Show that E(U) = ϵx.
(b) Show that Var(U) = (1 − )σ² + ϵx² − ϵ²x².
(c) Obtain the partial derivative of the right side of this last equation with respect to ϵ. This is the influence function.