firstly JakeD and topsquark i cant thank you enough

Top shark, yes its definitley an E

yes $\displaystyle T_j$ is assumed to be exponentially distributed

also yes this term does cancel out

and becomes

$\displaystyle P_n(z) = \prod^n_{j=2}\left(1-\frac{(z-1)\theta}{j-1}\right)^{-1} $

did he substitute a value for $\displaystyle T_j $?

i dont know why the original equation which is of the form of a poisson generating function becomes what is is now, which you said is a gamma(1,1).

im almost 100% certain this is not wrong, as its a very famous theory developed in mathematical biology and very well founded.

thank you so much for you help again its really much appreciated. If you want i can write the entire derivation our for you guys, will this help?