An Embedded Process

Suppose we have a birth and death process. The rate at which births occur isand the rate at which deaths occur isSometimes we are not interested the time of a particular process but only in whether it is a birth or a death. We can relabel the time of the ith change asand obtain a random processTime has been made discrete. The process is said to be embedded in the original process andis the size of the population immediately after the ith change.

For the simple birth and death process the probability of a birth isand the probability of a death is

This means that we have a simple random walk withand

If x ever becomes zero, it stays zero permanently and the population is extinct. The problem is now a gambler's ruin problem. Using results from the gambler's ruin problem we can say that if the initial population is x_0 then the probability of extinction isifand 1 if

The expected number of points (births and deaths) before the population goes extinct isifThe expected time to extinction is infinite if