Editing Multinomial distribution (section)

== Definitions ==
===Probability mass function===
Suppose one does an experiment of extracting ''n'' balls of ''k'' different colors from a bag, replacing the extracted balls after each draw. Balls of the same color are equivalent. Denote the variable which is the number of extracted balls of color ''i'' (''i'' = 1, ..., ''k'') as ''X''<sub>''i''</sub>, and denote as ''p''<sub>''i''</sub> the probability that a given extraction will be in color ''i''. The [[probability mass function]] of this multinomial distribution is:

: <math> \begin{align}
f(x_1,\ldots,x_k;n,p_1,\ldots,p_k) & {} = \Pr(X_1 = x_1 \text{ and } \dots \text{ and } X_k = x_k) \\
& {} = \begin{cases} { \displaystyle {n! \over x_1!\cdots x_k!}p_1^{x_1}\times\cdots\times p_k^{x_k}}, \quad &
\text{when } \sum_{i=1}^k x_i=n \\  \\
0 & \text{otherwise,} \end{cases}
\end{align}
</math>

for non-negative integers ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>.

The probability mass function can be expressed using the [[gamma function]] as:

:<math>f(x_1,\dots, x_{k}; p_1,\ldots, p_k) = \frac{\Gamma(\sum_i x_i + 1)}{\prod_i \Gamma(x_i+1)} \prod_{i=1}^k p_i^{x_i}.</math>

This form shows its resemblance to the [[Dirichlet distribution]], which is its [[conjugate prior]].

=== Example ===

Suppose that in a three-way election for a large country, candidate A received 20% of the votes, candidate B received 30% of the votes, and candidate C received 50% of the votes.  If six voters are selected randomly, what is the probability that there will be exactly one supporter for candidate A, two supporters for candidate B and three supporters for candidate C in the sample?

''Note: Since we’re assuming that the voting population is large, it is reasonable and permissible to think of the probabilities as unchanging once a voter is selected for the sample.  Technically speaking this is sampling without replacement, so the correct distribution is the [[Hypergeometric distribution#Multivariate hypergeometric distribution|multivariate hypergeometric distribution]], but the distributions converge as the population grows large in comparison to a fixed sample size''<ref>{{Cite web |title=probability - multinomial distribution sampling |url=https://stats.stackexchange.com/a/335239/307588 |access-date=2022-07-28 |website=Cross Validated |language=en}}</ref>''.''

: <math> \Pr(A=1,B=2,C=3) = \frac{6!}{1! 2! 3!}(0.2^1) (0.3^2) (0.5^3) = 0.135 </math>