Goodman and Kruskal XXXXXXXXXXproposed an association measure (tau) for nominal variables based on variation measure a. Show V(Y) is the probability that two independent observations on Y fall in...

Goodman and Kruskal (1954) proposed an association measure (tau) for nominal variables based on variation measure

a. Show V(Y) is the probability that two independent observations on Y fall in different categories (called the Gini concentration index).

Show that V(Y) = 0 when π_+j 1 for some j and V(Y) takes maximum value of (J — 1)/J when π_+j = 1/J for all j.

b. For the proportional reduction in variation, show that E[V(Y |X)] [The resulting measure (2.12) is called the concentration coefficient. Like U, T = 0 is equivalent to indepen­dence. Haberman (1982) presented generalized concentration and uncertainty coefficients.]