From multinomial distribution to binomial distribution

[binjip]

Homework Statement

(N₁, ... , N_r) has multinomial distribution with parameters n and p₁, ... , p_r.

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 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 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 N_i ) 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
N_i 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!/(N_i! * N_j! * (n - N_i - N_j)!) * p_i^N_i * p_j^N_j * (1-p_i-p_j)^(1-N_i-N_j)

Do you think this makes sense?

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

 

[mighty2000]

Your reasoning for a) and b) is correct. The "n" in Bin(n, p) refers to the number of trials, so in this case it would be 10.

For c), your formula is correct. The joint distribution of Ni, Nj, and n - Ni - Nj can be thought of as the probability of getting a specific combination of successes for each of the r categories. In this case, we are looking at the probability of getting Ni successes in category 1, Nj successes in category 2, and n - Ni - Nj successes in the remaining categories.

To gain more intuition, you can think of the joint distribution as a way to model the probability of getting a certain outcome in a game or experiment with multiple categories. For example, if you were rolling a die and wanted to know the probability of getting a 2 on the first roll, a 3 on the second roll, and a 5 on the third roll, you could use a multinomial distribution with r=3 (representing the 3 rolls) and p1=1/6, p2=1/6, p3=1/6 (representing the probabilities for each number on the die). The joint distribution would then give you the probability of getting that specific combination of outcomes.

In general, the multinomial distribution is useful for modeling situations where you have multiple categories and want to know the probability of a specific combination of outcomes.