perhaps i should rather ask
if i had
how would i remove the E
i should start by asking is anyone on the forum well versed in the coalescent theory?
i have a problem in a derivation
if the number of mutations is given by
as one can see this is the probability generating function of a poisson random variable with random mean. where
now from here i can write this as a product
so
Now im confused what to do. If i have the probability generating function how do i find the density function (derivative?)
the next step in the derivation should be
I cant for the life of me figure out how this was found from the pgf. PLEASE ANY IDEAS ARE WELCOME, SO IF NO ONE KNOWS THE ANSWER SUGGEST ANYTHING.
many thanks
chogo
thank you jakeD sorry yeah i should not have put the expectation there. My bad. Was just trying to write something simpler to see if anyone had any suggestions
The expectation is valid in my original equation, where the random variable is
the thing is how did the person who did that derivation remove the expectation? and arrive at the second formula.
->
any suggestions?
i thank you so much for your help
What is the distribution of the random variable You haven't said. The expectation uses that distribution.
Let be the moment generating function of Then comparing the first and second equations
where Then
which is the MGF of a Gamma(1,(j-1)/2j) distribution, that is, an exponential distribution with parameter So it appears that is the distribution of Is that correct?
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?
thank your help is invaluable. but one last problem. Like you before i used the definition
to go from
but when you said you compared the forms of the first and second equations to arrive at the third equation im not fully satistifed.
Is there a formal definition which says the moment generating function i have is equal to
where
I was trying to deduce what the distribution of was because you didn't say what it was. So I deduced it was an exponential distribution with parameter Is this correct?
Now you know what the distribution of is. So you can say what the moment generating function for is. I was working backwards to deduce the distribution; you can work forwards knowing the distribution.
I got that when from a probability text. See Exponential distribution - Wikipedia, the free encyclopedia .