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I was reading these notes on mathematical population genetics, and they have a derivation of the mean time ($\tau$) until either of the alleles fix, in the Wright-Fisher model. They get, in page 12, after equation (2.1.10):

$$E(\tau|H_0) = 2NH_0$$

where $H_0$ is the initial heterozygocity, defined there as:

$$H_n = \frac{2X_n (2N-X_n)}{2N(2N-1)}$$

although, sometimes it is defined instead as $H_n = \frac{2X_n (2N-X_n)}{(2N)^2}$. However, as said here, Kimura and Ohta showed

$$E(\tau|p)=-4N\cdot \left((1-p)\cdot \ln(1-p)+p\cdot \ln(p)\right)$$

where $p = X_0/N$ is the initial frequency of the allele.

So, I just want to check if the notes I'm reading are wrong, or I'm just missing something?

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  • $\begingroup$ The first formula is incorrect. $\endgroup$ – Artem Apr 14 '16 at 3:13
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    $\begingroup$ (2.1.10) relies on / is more or less equivalent to, the assertion that $$P(X_1\ne0,2N\mid X_0)=\frac{X_0(2N-X_0)}{2N^2}.$$ But the LHS is $$1-p(X_0,0)-p(X_0,2N)=1-\left(\frac{2N-X_0}{2N}\right)^{2N}-\left(\frac{X_0}{2N}\right)^{2N},$$ which is not the RHS in general. $\endgroup$ – Did Apr 17 '16 at 9:07

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