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