Question

Consider a Yule process with parameter \(\beta\) and initial state \(N=1\). Suppose the first individual is also subject to death, with the probability of death in the interval \(t\) to \(t+h\), given that the individual is living at time \(t\), being \(\mu h+o(h)\). Compute the distribution of the number of offspring due to a single individual and his descendants at the time of death of the original parent.

Short answer

The distribution of the number of offspring due to a single individual and his descendants at the time of death of the original parent is a mixture of Poisson distributions with the Gamma distribution as the mixing distribution. The parameters of the Gamma distribution are \(\beta\) and \(\mu\), where \(\beta\) is the birth rate and \(\mu\) is the rate of death.

Step-by-step solution

Understand the Yule process

A Yule process describes the growth of population when each individual in the population reproduces independently at a constant rate \(\beta\). It is a simple birth process where the inter-event times are independent Exponential random variables with rate \(\beta\). This implies that the number of offspring produced by an individual follows a Poisson process with parameter \(\beta\).

Include death process in the model

In addition to the Yule process, the first individual is also subject to death, with the probability of death in the interval \(t\) to \(t+h\) being \(\mu h+o(h)\). The death event of an individual can be modelled using an exponential distribution with rate \(\mu\). The random time \(T\) until the death of the individual is thus an Exponential random variable with parameter \(\mu\). Hence, the number of offspring when the original parent dies is a stopped Poisson process at the random time \(T\). It follows the Poisson distribution with the parameter equal to the product of the birth rate and the random time \(T\).

Compute the distribution of the number of offspring

The distribution of the number of offspring due to a single individual and his descendants at the time of death of the original parent is a mixture of Poisson distributions. The parameter of the Poisson distribution is the product of the birth rate and a random variable \(T\) that follows an Exponential distribution. This is a compound Poisson distribution, the expectation of which can be calculated by integrating the product of the birth rate and the density function of \(T\) over all possible values of \(T\). This results in a Gamma distribution with parameters \(\beta\) and \(\mu\).