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

\(X=(X_1,X_2,\ldots,X_k)\)

given the total

\(n=X_1+X_2+\ldots+X_k\)

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

\(\pi=(\pi_1,\pi_2,\ldots,\pi_k)\)

and

\(\pi_j=\dfrac{\lambda_j}{\lambda_1+\lambda_2+\cdots+\lambda_k}\)

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,

\(n=X_1+X_2+\cdots+X_7\)

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.