Mahler's notes question on adding compound distributions

[MarktheSharp]

In Mahler's section on Frequency Distributions under Section 12, he talks about adding compound distributions. He makes a statement about a primary distribution being infinitely divisible (such as Poisson or Negative Binomial) therefore we can add up these compound distributions. Then in one of his examples he uses a compound Binomial distribution and proceeds to add them, but I thought a Binomial was not infinitely divisible. I am a little confused and was wondering how we know which compound distributions can be added??

Any clarification would be great.

Thanks

[Howard Mahler]

One can add Binomials. One can not divide Binomials into infinitely smaller pieces.
So yes, one can also add compound distributions in which the primary distribution is Binomial.

From the Fall 2005 edition:

"Since the Poisson and the Negative Binomial are each infinitely divisible, so are compound distributions with a primary distribution which is either a Poisson or a Negative Binomial (including a Geometric.) Thus we can add up or thin either Compound Poisson or Compound Negative Binomial distributions."

"If one adds independent identically distributed Compound Binomial variables one gets the same form."

Howard

Please send me future questions via email.

Originally Posted by MarktheSharp

In Mahler's section on Frequency Distributions under Section 12, he talks about adding compound distributions. He makes a statement about a primary distribution being infinitely divisible (such as Poisson or Negative Binomial) therefore we can add up these compound distributions. Then in one of his examples he uses a compound Binomial distribution and proceeds to add them, but I thought a Binomial was not infinitely divisible. I am a little confused and was wondering how we know which compound distributions can be added??

Any clarification would be great.

Thanks