2.3.6 - Relationship between the Multinomial and the Poisson
2.3.6 - Relationship between the Multinomial and the PoissonSuppose that \(X_{1}, \dots, X_{k}\) 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
\(X=(X_1,\ldots,X_k)\)
given the total \(n=X_1+\ldots+X_k\) is \(Mult\left(n, \pi\right)\), where \(\pi=(\pi_1,\ldots,\pi_k)\), and
\(\pi_j=\dfrac{\lambda_j}{\lambda_1+\cdots+\lambda_k}\)
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 \(X = \left(X_{1}, \dots, X_{k}\right)\) given \(n\),
\(X\sim Mult(n,\pi)\)
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, and any estimates, tests, etc. for \(\pi\) or functions of \(\pi\) will be the same, whether we regard \(n\) as random or fixed.
Example: Vehicle Color
Suppose, while waiting at a busy intersection for one hour, we 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, \(n=X_1+\cdots+X_7\) 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, \(n\) is random.)
In this case, it's reasonable to regard the \(X_{j}\)s as independent Poisson random variables with means \(\lambda_{1}, \ldots, \lambda_{7}\). 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 sizes \(n_{j}\) as random or fixed. That is, we can proceed as if
\(X=(X_1,\ldots,X_7)\sim Mult(n,\pi)\)
where \(\pi = \left(\pi_{1}, \dots, \pi_{7}\right)\), even though \(n\) is actually random.