evolution - Heterozygosity under genetic drift

The wright-Fisher model of genetic drift is:

$$p_{ij} = \binom{2N}{j}\left(\frac{i}{2N}\right)^j \left(1- \frac{i}{2N}\right)^{2N-j} $$

,where $\binom{2N}{j}$ is a binomial coefficient.

From this equation one can infer that the expected heterozygostiy should decrease by $1-\frac{1}{2N}$ at each time step because:

$$E[x_{t+1}(1-x_{t+1}) \space|\space x_t] = (1-\frac{1}{2N})x_t(1-x_t)$$

I don't understand this equality. That might be very simple though! Can you help me making sense of it?

source

Answer

To derive it, first use that $E[x(1-x)]= E[x-x^2]=E[x]-E[x^2]$ and that $E[x^2]=\text{Var}[x]+E[x]^2$ to rewrite the left-hand side: $$E\left[x_{t+1}(1-x_{t+1})\right] = E\left[x_{t+1}\right](1-E\left[x_{t+1}\right])-\text{Var}\left[x_{t+1}\right].$$ The equation for $p_{ij}$ is just saying that $2Nx_{t+1}$ is binomially distributed with $2N$ trials with success probability $x_t$, so $E[x_{t+1}]=x_t$ and $\text{Var}[x_{t+1}]=x_t(1-x_t)/(2N)$. (Note that $\text{Var}[\alpha x]=\alpha^2\text{Var}[x]$.) Plugging these values into the equation above gives the form you're looking for.