The Most Recent Common Ancestor of a Randomly Chosen Sample in a Galton-Watson Process

Abstract

One of the interesting topics to be examined in the eld of theoretical population genetics is to nd the most recent common ancestor of the current population. In this thesis, rstly we proceed by nding the common ancestor of two randomly chosen individuals at the current time s, thus we will build a mathematical model to illustrate this situation using a Galton-Watson process which allows the pop- ulation to grow stochastically over time, and only genetic drift a ects the model. We then generalise the model into modelling the common ancestor for n ran- domly chosen individuals at the current time step. Later, we derive the joint estimate of having a common ancestor from n chosen individuals and the current population at time s does not exceed the observed population. Hence, this will be used to determine the estimate of the initial population size and the time of the most recent common ancestor for the case of n = 2. Lastly, we will compare the coalescent time with result found by Slatkin and Hudson (1991) using the Wright-Fisher model.