Let Z1, Z2, . . . , ZN describe a branching process in which each parent has j offspring with probability pj . Find the probability d that the process eventually dies out if
(a) p0 = 1/2, p1 = 1/4, and p2 = 1/4.
(b) p0 = 1/3, p1 = 1/3, and p2 = 1/3.
(c) p0 = 1/3, p1 = 0, and p2 = 2/3.
(d) pj = 1/2j+1, for j = 0, 1, 2, . . . .
(e) pj = (1/3)(2/3)j , for j = 0, 1, 2, . . . .
(f) pj = e−22j/j!, for j = 0, 1, 2, . . . (estimate d numerically).