I'm currently writing a set of notes on coalescent theory and I'm writing an example on how to directly calculate the probability of a given rooted genealogical tree. This is done by writing an expression for the probability in terms of the waiting time and then solving that. We have that
$T_n \sim $ Exp$\left(\binom{n}{2}\right)$
And the expression I need to take the expectation of is
$\frac{\theta^4}{2^4}\left(e^{-\theta\frac{2T_4 + 3T_3 + 2T_2}{2}}\left(T_3T_2(T_4+T_3)(T_4+T_3+T_2)\right) \right)$
Where $\theta$ is some constant representing the mutation rate.
I'm totally lost on how to calculate this directly. A solution or resources on how to do this myself would be very much appreciated.
Thank you
If $T$ is exponential with mean $m$ then $X=T/m$ is exponential with mean 1. Therefore your complicated exponential can be written $e^{aX_1+bX_2+cX_3}$ and its expectation is $\frac{1}{(1-a)(1-b)(1-c)}. $ You are interested in the expectation of the product of this exponential by a complicated polynomial, therefore enough is to compute for suitable integers $\alpha,\beta,\gamma$
$$E(e^{aX_1+bX_2+cX_3}X_1^{\alpha}X_2^{\beta}X_3^{\gamma})$$$$=(\frac{d}{da})^{\alpha}\frac{1}{1-a}\times (\frac{d}{db})^{\beta}\frac{1}{1-b}\times (\frac{d}{dc})^{\gamma}\frac{1}{1-c}$$$$=\frac{\alpha!}{(1-a)^{\alpha+1}}\times \frac{(\beta!}{(1-b)^{\beta+1}}\times \frac{\gamma!}{(1-c)^{\gamma+1}}$$