1.7.7 - Relationship between the Multinomial and Poisson

Suppose that \(X_{1}, X_{2}, \dots, X_{k}\) are independent Poisson random variables,

\(\begin{aligned}&X_{1} \sim P\left(\lambda_{1}\right)\\&X_{2} \sim P\left(\lambda_{2}\right)\\&...\\&X_{k} \sim P\left(\lambda_{k}\right)\end{aligned}\)

where the \(\lambda_{j}\)'s are not necessarily equal. Then the conditional distribution of the vector


given the total


is \(\text{Mult}\left(n, \pi\right)\), where




That is, \(\pi\) is simply the vector of \(\lambda_{j}\)'s normalized to sum to one.

This fact is important, because it implies that the unconditional distribution of \(\left(X_{1}, \dots, X_{k}\right)\) can be factored into the product of two distributions: a Poisson distribution for the overall total,

\(n\sim P(\lambda_1+\lambda_2+\cdots+\lambda_k)\)

and a multinomial distribution for \(X = \left(X_{1}, X_{2}, \dots, X_{k}\right)\) given n,

\(X\sim \text{Mult}(n,\pi)\)

The likelihood factors into two independent functions, one for \(\sum\limits_{j=1}^k \lambda_j\) and the other for \(\pi\). The total n carries no information about \(\pi\) and vice-versa.

Therefore, likelihood-based inferences about \(\pi\) are the same whether we regard \(X_{1}, \dots, X_{k}\) as sampled from k independent Poissons or from a single multinomial. That is, any estimates, tests, etc. for \(\pi\) or functions of \(\pi\) will be the same whether we regard n as random or fixed.

Example 1-11 Section

Suppose that you wait at a busy intersection for one hour and record the color of each vehicle as it drives by. Let

\(X_{1} =\) number of white vehicles

\(X_{2} =\) number of black vehicles

\(X_{3} =\) number of silver vehicles

\(X_{4} =\) number of red vehicles

\(X_{5} =\) number of blue vehicles

\(X_{6} =\) number of green vehicles

\(X_{7} =\) number of any other color

In this experiment, the total number of vehicles observed,


is random. (It would have been fixed if, for example, we had decided to classify the first n = 500 vehicles we see. But because we decided to wait for one hour, the n is random.)

In this case, it's reasonable to regard the \(X_{j}\)'s as independent Poisson random variables with means \(\lambda_{1}, \lambda_{2}, \dots, \lambda_{7}\). But if our interest lies not in the \(\lambda_{j}\)'s but in the proportions of various colors in the vehicle population, inferences about these proportions will be the same whether we regard the sample size \(n_{j}\)'s as random or fixed. That is, we can proceed as if

\(X=(X_1,\ldots,X_7)\sim \text{Mult}(n,\pi)\)

where \(\pi = \left(\pi_{1}, \dots, \pi_{7}\right)\), even though n is actually random.