# From multinomial distribution to binomial distribution

• binjip
In summary: So, in summary, the (N1, ... , Nr) has multinomial distribution with parameters n and p1, ... , pr and the joint distribution of Ni, Nj, and n - Ni - Nj can be calculated using the multinomial distribution function. This is useful for modeling situations where there are multiple categories and you want to know the probability of a specific combination of outcomes.
binjip

## Homework Statement

(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

## 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 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:

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.

## 1. What is a multinomial distribution?

A multinomial distribution is a probability distribution that describes the frequency of each possible outcome in a multi-category experiment. It is used to model situations where there are more than two possible outcomes for each trial, such as rolling a die or drawing a colored ball from a bag.

## 2. How is a multinomial distribution related to a binomial distribution?

A multinomial distribution can be thought of as multiple binomial distributions combined. In a binomial distribution, there are two possible outcomes (success or failure) for each trial. In a multinomial distribution, there are more than two possible outcomes for each trial, but the probabilities of success and failure are still fixed.

## 3. What is the difference between a multinomial distribution and a binomial distribution?

The main difference between a multinomial distribution and a binomial distribution is the number of possible outcomes for each trial. A binomial distribution has only two possible outcomes, while a multinomial distribution has more than two possible outcomes. Additionally, in a binomial distribution, the probabilities for each trial are the same, while in a multinomial distribution, the probabilities for each outcome can vary.

## 4. How do you calculate probabilities in a multinomial distribution?

In a multinomial distribution, the probabilities for each outcome are calculated using the formula P(x1, x2, ..., xk) = (n!/(x1!x2!...xk!)) * (p1^x1 * p2^x2 * ... * pk^xk), where n is the number of trials, x1, x2, ..., xk are the number of successes for each outcome, and p1, p2, ..., pk are the probabilities of success for each outcome.

## 5. What are some real-world applications of multinomial distributions?

Multinomial distributions are commonly used in fields such as genetics, market research, and social sciences. They can be used to model the outcomes of genetic crosses, consumer purchasing behavior, and voting patterns, among others. They are also used in risk analysis and decision making to assess the likelihood of multiple outcomes occurring simultaneously.

• Calculus and Beyond Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
13
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Quantum Interpretations and Foundations
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
10
Views
2K
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
3K
• Calculus and Beyond Homework Help
Replies
1
Views
959