firstly JakeD and topsquark i cant thank you enough

Top shark, yes its definitley an E

yes

is assumed to be exponentially distributed

also yes this term does cancel out

and becomes

did he substitute a value for

?

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?