=+24. Continuing Problem 22, consider a single-type process starting from a single particle. Suppose that the progeny generating function is

∞

d=0 pdzd = zk for some k ≥ 2 and the death intensity is λ. Show that the ordinary differential equation

∂

∂tP(t, z) = −λP(t, z) + λP(t, z)

k for the generating function of Xt has solution P(t, z) = ze−λt

1 − zk−1 + e−λ(k−1)t zk−1

−1/(k−1)

.

Also check the initial condition P(0, z) = z. What is P(t, z) if you start from j > 1 particles?