1
$\begingroup$

I was looking at the properties of the Binomial probability distribution and it says that the variance is np(1-p). In population genetics, n = 2N. So I would expect to see that the variance is 2Np(1-p).

But when looking at the Wright-Fisher model, I often see that the variance is $p(1-p)/(2N)$ (see this presentation page 6).

How to derive the variance of the Wright-Fisher model to get this variance from $Prob\{X=i\}=\frac{n!}{i!(n-i)!}p^i(1-p)^{n-i}$?

$\endgroup$
1
$\begingroup$

The binomial variance $2N p (1-p)$ is for the number of individuals $n'$ carrying the allele in the next generation. The frequency of the allele in the next generation is $p'=n'/(2N)$, so its variance is $$\text{Var}[p'] = \text{Var}[n'/(2N)] = \text{Var}[n']/(2N)^2 = p(1-p)/(2N).$$

$\endgroup$
  • $\begingroup$ This is because $Var[n'] = 2Npq$ which is the variance of the binomial model $(2Np(1-p))$ Right? Because $n'$ is also coming from a binomial sampling $\endgroup$ – M. Beausoleil Aug 16 at 14:02
  • 1
    $\begingroup$ @M.Beausoleil: yes, exactly. $\endgroup$ – Daniel Weissman Aug 17 at 17:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.