Mean

One useful property to know is the expected value of \(X_t\) at any time throughout the process. This can be computed using the fact that the expectation of a binomial distribution is \(np\) and the law of total expectation:

\[E(X_t) = E(E(X_t \mid X_{t-1})) = E(2N \frac{X_{t-1}}{2N})\] \[= E(X_{t-1}) = E(E(X_{t-1} \mid X_{t-2})) = E(2N \frac{X_{t-2}}{2N})\] \[= E[X_{t-2}] = \dots = X_0\]

We see that the expected value of the Wright-Fisher process at any time-point is just the count of the \(A\) allele in the first generation. If \(Y_t\) is the frequency of the \(A\) allele at time \(t\) then the above result can be extended such that:

\[E(Y_t) = Y_0\]

Variance

It is a little bit harder to find the variance of \(X_t\) but it is useful to know and will be important for later tutorials. Recall from basic probability theory the definition of the variance:

\[Var(X_t) = E(X^2_t) - E(X_t)^2\]

We have shown about that \(E(X_t) = X_0\) thus:

\[Var(X_t) = E(X^2_t) - X^2_0\]

Using the law of total variance and the mean and variance of the binomial distribution:

\[E(X^2_{t}) = E(E(X^2_{t} \mid X_{t-1})) = E(Var(X_{t} \mid X_{t-1}) + E(X_{t} \mid X_{t-1})^2)\] \[ = E(2N\frac{X_{t-1}}{2N}(1 - \frac{X_{t-1}}{2N}) + (2N\frac{X_{t-1}}{2N})^2) \] \[ = E(X_{t-1} - \frac{X^2_{t-1}}{2N} + X^2_{t-1}) = E(X^2_{t-1} - \frac{X^2_{t-1}}{2N} + X_{t-1})\] \[= E(X^2_{t-1}) (1 - \frac{1}{2N}) + E(X_{t-1})\]

thus we can rewrite \(E(X^2_{t})\) as:

\[Var(X_{t}) + X^2_0 = (Var(X_{t-1}) + X^2_0) (1 - \frac{1}{2N}) + X_0\] \[= Var(X_{t-1})(1 - \frac{1}{2N}) + X^2_0 - \frac{X^2_0}{2N} + X_0\] \[\Rightarrow Var(X_{t}) = Var(X_{t-1})(1 - \frac{1}{2N}) + X_0(1 - \frac{X_0}{2N})\]

solving the above recurrence results in:

\[Var(X_{t}) = 2N X_0 (1 - \frac{X_0}{2N}) (1 - (1 - \frac{1}{2N})^t)\]

and finally in terms of the frequency of the \(A\) allele:

\[Var(Y_{t}) = Y_0 (1 - Y_0) (1 - (1 - \frac{1}{2N})^t)\]

Fixation probabilities

Using the above moments we can compute some interesting features of the Markov chain which also have important biological relevance. Recall that absorbing states of a Markov chain are the states, once entered, can never be exited. More formally, if \(X^{*}_t\) is an absorbing state then:

\[P_{\{i = {X^{*}_t}\}j} = 0\]

In the Wright-Fisher model we can intuitively see that the fixation or loss of an allele are absorbing states i.e. if all of the individuals in the population carry two copies of the \(A\) allele or \(a\) allele the allele of the other type cannot be sampled without mutation or migration. The absorbing states in the Wright-Fisher model are \(X^{*}_t \in \{0, 2N\}\). We can compute the probability of fixation of the \(A\) allele, transitioning to the absorbing state \(X^{*}_{t+1} = 2N\) from any other state, using the conditional expectation described above:

\[E(X^{*}_{t+1} \mid X_t = i) = 2NP(X_{t+1} = 2N \mid X_t = i) \] \[\Rightarrow i = 2NP(X_{t+1} = 2N \mid X_{t} = i)\] \[\Rightarrow P(X_{t+1} = 2N \mid X_{t} = i) = \frac{i}{2N}\]

Therefore we can reciprocally write the probability of losing the \(A\) allele (fixation of the \(a\) allele):

\[ P(X_{t+1} = 0 \mid X_{t} = i) = 1 - P(X_{t+1} = 2N \mid X_{t} = i) = 1 - \frac{i}{2N}\]

We can see that the probability of fixation or loss of an allele in a pure drift Wright-Fisher model only depends on the previous count of the allele and the effective population size.