where m = m + , m 1, and , > 0. Show, subject to suitable extra conditions, that pm(t) = P(M(t )

where λm = αm + β, m ≥ 1, and α, β > 0. Show, subject to suitable extra conditions, that pm(t) = P(M(t ) = m) satisfies a set of differential–difference equations to be specified. Deduce without solving these equations in their entirety that M(t ) has mean

β(eαt − 1)/α, and find the variance of M(t).

(Cambridge 2001)