I'm trying to follow the maths of a population genetics paper. Basically, I am stuck in a manipulation where the authors notice that a given function has the shape of a chi square distribution and they transform it in a function with two gamma functions.
The equation is the following:
$$\ln(\bar{W})= –\frac{2V_g^{Q/2}}{2V_s} E\sum^n_{i=1}\left(\left(\frac{z_i}{\sqrt{v_g}}\right)^2\right)^{Q/2}$$
which becomes
$$\ln(\bar{W})= –\frac{(2V_g)^{Q/2}}{2V_s}\left(\frac{\Lambda{\frac{n+Q}{2}}}{\Lambda{\frac{n}{2}}}\right)$$
I would like to understand this transformation.
Cheers, João
The placement of the parentheses in the expression is incorrect. Here is what the problem should be.
$\{z_i\}$ are iid normal with mean $0$ and variance $v_g.$ We want to evaluate
$$\ln(\bar{W})= –\frac{1}{2v_s} E\left[ \left(\sum^n_{i=1} z_i^2 \right)^{Q/2}\right]$$
$$\ln(\bar{W})= –\frac{v_g^{Q/2}}{2v_s} E\left[ \left(\sum^n_{i=1} \frac {z_i^2}{v_g} \right)^{Q/2}\right]$$
Then $ \sum^n_{i=1} \frac {z_i^2}{v_g} $ is the sum of the squares of independent standard normal rv. This sum has a chi-square distribution with $n$ df. Now we need the $Q/2$ moment of such a rv. That happens to be:
$$\frac{\Gamma(\frac{n+Q}{2}) 2^{Q/2} } {\Gamma({\frac{n}{2} ) }}$$
which yields $$\ln(\bar{W})= –\frac{(2v_g)^{Q/2}}{2v_s}\frac{\Gamma(\frac{n+Q}{2}) } {\Gamma({\frac{n}{2} ) }}$$