Editing Multinomial distribution (section)

===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]].