Von Neumann’s method for the exponential distribution has been extended in some interesting ways, two of which show some remarkable improvement in speed. In the first one (due to Marsaglia 1961), let...

Von Neumann’s method for the exponential distribution has been extended in some interesting ways, two of which show some remarkable improvement in speed. In the first one (due to Marsaglia 1961), let M have the geometric distribution with parameter e−1 , Pr(M = m) = (1−e−1 )e−m, and let N have the Poisson distribution with zero removed: Pr(N = n) = 1/[n!(e − 1)]. Show that X = M + min(U1,...,UN ) has the exponential distribution. Following this result, write an algorithm using inversion for the two discrete distributions and compare it to an implementation of the von Neumann algorithm. Sibuya (1962) generalized to M geometric with Pr(M = m) = (eµ −1)/eµ(m+1) and well as N (with the zero removed) Poisson with parameter µ, Pr(N = n) = µn /(eµ − 1)n!. Show that this also will lead to X having the exponential distribution. Implement Sibuya’s method with µ = log 2, so that the geometric is a shift count, and compare it to the other two.