# From multinomial distribution to binomial distribution

1. Jul 3, 2011

### binjip

1. The problem statement, all variables and given/known data

(N1, ... , Nr) has multinomial distribution with parameters n and p1, ... , pr.

Let 1 $\leq$ i < j $\leq$ r.

I am looking for an intuitive explanation for the 3 following questions.

a) What is the distribution of Ni?
b) What is the distribution of Ni + Nj?
c) What is the joint distribution of Ni, Nj, and n - Ni - Nj

2. Relevant equations

Binomial and multinomial distribution function.

3. The attempt at a solution

An example of multinomial distribution with p1 = 0.2, p2 = 0.3, p3 = 0.5
P(N1=2, N2=3, N3=5) = $\frac{10!}{2!*3!*5!}$ * 0.22 * 0.33 * 0.55 = 0.08505

a) I believe the distribution of N1 (or any other Ni ) is Bin(n, p1). In this particular case, this would be the probability of having 2 successes in 10 trials.

Is my reasoning correct? I'm especially in doubt about the "n" in Bin(n, ...).

b) If a) is correct
Ni has Bin(n, p1) distribution.

from the concrete example, e.g. P(N1 + N2 = 5) = 10! / (5!(10-5)!) * (0.2 + 0.3)5 = 0.5(10-5) = 0.246

This is the probability of having 5 successes in 10 trials, but we don't care if the success comes from N1 or N2, hence the probabilities are summed.

The distribution of N1 + N2 becomes Bin(n, p1 + p2).

c) The joint distribution is multinomially distributed with

n!/(Ni! * Nj! * (n - Ni - Nj)!) * piNi * pjNj * (1-pi-pj)(1-Ni-Nj)

Do you think this makes sense?

Now any additional intuition and/or mathematical explanation behind this problem would be appreciated. Many thanks.

Last edited: Jul 3, 2011
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