Consider the Gini index θ (F) as defined. (a) Suppose that X ∼ F and let G be the...

Consider the Gini index θ_{(F)} as defined.

(a) Suppose that X ∼ F and let G be the distribution function of Y = a_{X} for some a > 0. Show that θ_{(G)} = θ_{(F)}.

(b) Suppose that F_{p} is a discrete distribution with probability p at 0 and probability 1 − p at x > 0. Show that θ(F_{p}) → 0 as p → 0 and θ(F_{p}) → 1 as p → 1.

(c) Suppose that F is a Pareto distribution whose density is

α > 0. (This is sometimes used as a model for incomes exceeding a threshold x_{0}.) Show that θ_{(F)} = (2α − 1)^{−1} for α > 1. (f(x; α) is a density for α > 0 but for α ≤ 1, the expected value is infinite.)