The gory details⁶

Starting where we left off above:

$$\begin{eqnarray*} \mbox{E}(\tilde H) &=& 2\left((\mbox{E}\hat p) - \mbox{E}({\ha... ... \\ &=& 2\left(p - \mbox{E}\left((k/n)^2\right)\right) \quad , \end{eqnarray*}$$

where $k$ is the number of $A_1$ alleles in our sample and $n$ is the sample size.

$$\begin{eqnarray*} \mbox{E}\left((k/n)^2\right) &=& \sum (k/n)^2 \mbox{P}(k) \\ ... ...+ p^2\right) \\ &=& (1/n)^2 \left(np(1-p) + p^2\right) \quad . \end{eqnarray*}$$

Substituting this back into the equation above yields the following:

$$\begin{eqnarray*} \mbox{E}(\tilde H) &=& 2\left(p - (1/n)^2 \left(np(1-p) + p^2\... ...left(1 - 1/n\right)2p(1-p) \\ &=& ((n-1)/n)2p(1-p) \quad . \\ \end{eqnarray*}$$