I am working with the distribution of the sum of two dependent random variables. In my problem, there are two unobserved events, X and Y, where X precedes Y and Y is a function of the outcome of X, but I only observe the sum of the two outcomes, i.e:
It looks like a standard convolution distribution, except that Y and X are not independent. In my particular case, X and Y have a Poisson distribution, where the mean of Y is an increasing function of the value of the outcome of X. I am unaware of a closed form for the representation of the PMF of this distribution, and so need to derive it directly through the above sum. For my particular problem, I need to compute the PMF thousands of times, and am looking for ways to reduce the computational burden of doing so. As such, I was hoping someone would either
1) Know if there is a closed form representation for the PMF
2) Know of any approximations to the PMF
Any help would be greatly appreciated.