Recall Banach’s matchbox problem from Example 4.13.

(a) One natural generalization is to allow for different numbers of matches in the left and right pockets, say N1 and N2, and probability p not necessarily 1/2 of drawing from the left side.14 Derive the mass function f (k; N1, N2, p)

and construct a program, say banach(n1,n2,p,sim) which computes it, simulates the process sim times, and finally plots the true and simulated mass functions overlaid. As an example, Figure 4.10

(a) was produced with vec=banach(30,10,1/2,10000);

text(3,0.07,’N_1=30, N_2=10, p=1/2’,’fontsize’,14)

(b) Further extensions could allow for more than two matchboxes; see, for example, Cacoullos (1989, p. 80(317,318)) for some analytic results. For now, assume that Banach has n ≥ 2 matchboxes distributed throughout his many pockets, each of which initially contains Ni > 0 matches, i = 1,...,n.

Associated with each box is a probability pi, with i pi = 1. Construct a program which simulates the process and, once an empty box is discovered, reports the minimum and maximum number of matches in the remaining matchboxes. Simulate a large number of times and plot the marginal mass functions together. For example, Figure 4.11 shows the result with N = [20, 50, 100], p = [0.1, 0.4, 0.5] and 20 000 replications.