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binjip

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## Homework Statement

(N

_{1}, ... , N

_{r}) has multinomial distribution with parameters n and p

_{1}, ... , p

_{r}.

Let 1 [itex]\leq[/itex] i < j [itex]\leq[/itex] r.

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

a) What is the distribution of N

_{i}?

b) What is the distribution of N

_{i}+ N

_{j}?

c) What is the joint distribution of N

_{i}, N

_{j}, and n - N

_{i}- N

_{j}

## Homework Equations

Binomial and multinomial distribution function.

## The Attempt at a Solution

An example of multinomial distribution with p

_{1}= 0.2, p

_{2}= 0.3, p

_{3}= 0.5

P(N

_{1}=2, N

_{2}=3, N

_{3}=5) = [itex]\frac{10!}{2!*3!*5!}[/itex] * 0.2

^{2}* 0.3

^{3}* 0.5

^{5}= 0.08505

a) I believe the distribution of N

_{1}(or any other N

_{i}) is Bin(n, p

_{1}). 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

N

_{i}has Bin(n, p

_{1}) distribution.

from the concrete example, e.g. P(N

_{1}+ N

_{2}= 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 N

_{1}or N

_{2}, hence the probabilities are summed.

The distribution of N

_{1}+ N

_{2}becomes Bin(n, p

_{1}+ p

_{2}).c) The joint distribution is multinomially distributed with

n!/(N

_{i}! * N

_{j}! * (n - N

_{i}- N

_{j})!) * p

_{i}

^{Ni}* p

_{j}

^{Nj}* (1-p

_{i}-p

_{j})

^{(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.

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