Simulating the outcome of a birth and death process

A method is given for sampling the outcome population size of a birth and death process running over a specific duration of time for an arbitrary starting population size. This is done by treating two cases depending on the size of the starting population. If the starting population size is small, then a sample outcome can be constructed by repeated sampling from the outcome distribution for starting size one; this being a computationally manageable distribution. For large starting sizes a sample outcome is gotten by sampling from the normal distribution which approximates the birth and death distribution. It is possible to find this normal distribution since, even for large starting sizes, the first and second moments of the birth and death process are readily computed.