# 1.7.7 - Relationship between the Multinomial and Poisson

Suppose that *X*_{1}, *X*_{2}, . . .,* X _{k}* are independent Poisson random variables,

X_{1}∼ P(λ_{1}),

X_{2}∼ P(λ_{2}),

...

X∼ P(λ_{k}),_{k}

where the λ* _{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 Mult(n, π), where

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

and

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

That is, π is simply the vector of λ_{j}'s normalized to sum to one.

This fact is important, because it implies that the unconditional distribution of (*X*_{1}, . . . , *X _{k}*) 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* = (*X*_{1},* X*_{2}, . . . , *X _{k}*) 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 π. The total *n* carries no information about π and vice-versa.

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

#### Example

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 λ

_{1}, λ

_{2}, . . . , λ

_{7}. But if our interest lies not in the λ

*'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*

_{j}*n*as random or fixed. That is, we can proceed as if

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

where π = (π_{1}, . . . , π_{7}), even though *n* is actually random.