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

(a) Generate two independent observations U1 and U2 from U[0, 1].

(b) Obtain T = αU1 + (α − αU1 )U2.

(c) Let X = U1 and Y = logα[T/(T + (1 − α)U2)], where α > 0,α = 1. (i) Show that the bivariate cdf of Frank’s distribution has the following form:

Hα(x,y) = P(X ≤ x,Y ≤ y) = logα



1 + (αx − 1)(αy − 1)

α − 1



.

(ii) Generate one observation of(X,Y ) based on two independent observations from U[0, 1]: 0.548291 and 0.179112. Use α = 4.