Multinomial Distribution

Properties

A multinomial experiment is a statistical experiment that has the following properties:

• The experiment consists of n repeated trials.
• Each trial has a discrete number of possible outcomes.
• On any given trial, the probability that a particular outcome will occur is constant.
• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

Note: A binomial experiment is a special case of a multinomial experiment. Here is the main difference. With a binomial experiment, each trial can result in two - and only two - possible outcomes. With a multinomial experiment, each trial can have two or more possible outcomes.

Theory

In a multinomial distribution, each trial results in exactly one of some fixed finite number k possible outcomes, with probabilities p1, ..., pk (so that pi ≥ 0 for i = 1, ..., k and ), and there are n independent trials.

Then if the random variables Xi indicate the number of times outcome number i is observed over the n trials,

the vector X = (X1, ..., Xk) follows a multinomial distribution with parameters n and p, where p = (p1, ..., pk).

Formula

Suppose one does an experiment of extracting n balls of k different colours from a bag, replacing the extracted ball after each draw. Balls from the same colour are equivalent.

Xi: number of extracted balls of colour i (i = 1, ..., k)

pi: probability that a given extraction will be in colour i.

Let there be n balls extracted.

The probability mass function of this multinomial distribution, that is, probability that X1 (ball of color 1) is extracted x1 times, X2 (ball of color 2) is extracted x2 times,........ is

Example

Suppose we have a bowl with 10 marbles - 2 red marbles, 3 green marbles, and 5 blue marbles. We randomly select 4 marbles from the bowl, with replacement. What is the probability of selecting 2 green marbles and 2 blue marbles?

To solve this problem, we apply the multinomial formula. We know the following:

The experiment consists of 4 trials, so n = 4.

The 4 trials produce 0 red marbles, 2 green marbles, and 2 blue marbles; so nred = 0, ngreen = 2, and nblue = 2.

On any particular trial, the probability of drawing a red, green, or blue marble is 0.2, 0.3, and 0.5, respectively.

Thus, pred = 0.2, pgreen = 0.3, and pblue = 0.5

We plug these inputs into the multinomial formula, as shown below:

P = [ n! / ( n1! n2! ... nk! ) ] ( p1n1 p2n2 . . . pknk )

P = [ 4! / ( 0! 2! 2! ) ] [ (0.2)0 (0.3)2 * (0.5)2 ] P = 0.135

Thus, if we draw 4 marbles with replacement from the bowl, the probability of drawing 0 red marbles, 2 green marbles, and 2 blue marbles is 0.135.

References: Multinomial distribution