Hello, I am doing an introductional statistics unit in my first semester at uni the moment. I have a question to do and I have tried it for several days, but I still did not solve it.

I have to derive a marginal distribution function for X. X is defined as the sum of N independent Bernoulli trials. N is a random variable that behaves according to a Poisson distribution. p, Lambda etc is not given, so I have to derive a general function.

Have not check your whole calculation, but one reminder for you from the first line:

You have correctly stated that in the summation, as required in a Binomial model. Note that you are calculating the pmf

and from the whole calculation process is fixed from the very beginning. (And if this calculation results holds for all support point of , then you have finished calculating the whole pmf of )

For each fixed , we have

when

Therefore all the initial terms from to vanished, and you are actually summing from .