
Sum of Binomial and Poisson
Simon de Visscher <sdevisscher@hotmail.com> wrote: Let be independent of . We seek the distribution of the sum . Solution: Let denote the joint pmf of :
Let and . Then the joint pmf of , say , is given by mathStatica's Transform function as:
Deriving the domain of support of and is a bit more tricky. The following diagram plots the space in the plane where .
Then the domain of support (the shaded region in the figure) can be defined as follows:
When : The pmf of is then obtained by summing out in each part of the domain: * If , the pmf of the sum is:
* If , the pmf of the sum is:
where Hypergeometric1F1[a, b, z] denotes the Kummer confluent hypergeometric function .
