10.4.9 Genest (1987) provides the following algorithm for generating random Samples from the so-called Frank family of bivariate distributions:

(a) Generate two independent observations U, and U2 from U[O, 11.

(b) Obtain T = aul +(a-au1)U2.

(c) Let X = U, and Y = log,[T/(T + (1 - a)&) ]w, here a > 0, a # 1. (i) Show that the bivariate cdf of Frank’s distribution has the following form:(ii) Generate one observation of ( X , Y ) based on two independent observations from U[O, 11: 0.548291 and 0.179112. Use a = 4.

Ha(x, y) = = P(X, Y y) = loga 1+ {1 (a1)(a1)] a-1