We consider a remarkable property of the Poisson distribution that has a connection to the multinomial distribution. We start with the following examples.

**Example 1**

Suppose that the arrivals of customers in a gift shop at an airport follow a Poisson distribution with a mean of per 10 minutes. Furthermore, suppose that each arrival can be classified into one of three distinct types – type 1 (no purchase), type 2 (purchase under $20), and type 3 (purchase over $20). Records show that about 25% of the customers are of type 1. The percentages of type 2 and type 3 are 60% and 15%, respectively. What is the probability distribution of the number of customers per hour of each type?

**Example 2**

Roll a fair die times where is random and follows a Poisson distribution with parameter . For each , let be the number of times the upside of the die is . What is the probability distribution of each ? What is the joint distribution of ?

In Example 1, the stream of customers arrive according to a Poisson distribution. It can be shown that the stream of each type of customers also has a Poisson distribution. One way to view this example is that we can split the Poisson distribution into three Poisson distributions.

Example 2 also describes a splitting process, i.e. splitting a Poisson variable into 6 different Poisson variables. We can also view Example 2 as a multinomial distribution where the number of trials is not fixed but is random and follows a Poisson distribution. If the number of rolls of the die is fixed in Example 2 (say 10), then each would be a binomial distribution. Yet, with the number of trials being Poisson, each has a Poisson distribution with mean . In this post, we describe this Poisson splitting process in terms of a “random” multinomial distribution (the view point of Example 2).

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Suppose we have a multinomial experiment with parameters , , , where

- is the number of multinomial trials,
- is the number of distinct possible outcomes in each trial (type 1 through type ),
- the are the probabilities of the possible outcomes in each trial.

Suppose that follows a Poisson distribution with parameter . For each , let be the number of occurrences of the type of outcomes in the trials. Then are mutually independent Poisson random variables with parameters , respectively.

The variables have a multinomial distribution and their joint probability function is:

where are nonnegative integers such that .

Since the total number of multinomial trials is not fixed and is random, is not the end of the story. The probability in is only a conditional probability. The following is the joint probability function of :

To obtain the marginal probability function of , , we sum out the other variables () in and obtain the following:

Thus we can conclude that , , has a Poisson distribution with parameter . Furrthermore, the joint probability function of is the product of the marginal probability functions. Thus we can conclude that are mutually independent.

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**Example 1**

Let be the number of customers per hour of type 1, type 2, and type 3, respectively. Here, we attempt to split a Poisson distribution with mean 30 per hour (based on 5 per 10 minutes). Thus are mutually independent Poisson variables with means , , , respectively.

**Example 2**

As indicated earlier, each , , has a Poisson distribution with mean . According to , the joint probability function of is simply the product of the six marginal Poisson probability functions.

I think your expositions, those I looked at, are good.

I have a problem which seems like it’s easy but my brain refuses to come through. Would you be willing to analyze a simple recursive algorithm that produces a distribution that seems to have order statistics of the Beta distribution; more or less?

It might serve as an non-obvious example.

I’m sorry for putting this on the wrong page. I thought I was on the “About” page when posting.

Ray

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