The multinomial distribution is a generalization of the binomial
distribution to multiple categories. It is perhaps easiest to think
that we first extend a `Bernoulli()`

distribution to include more
than two categories, resulting in a `Categorical()`

distribution.
We then extend repeat the Categorical experiment several (\(n\))
times.

Multinomial(size, p)

## Arguments

size |
The number of trials. Must be an integer greater than or equal
to one. When `size = 1L` , the Multinomial distribution reduces to the
categorical distribution (also called the discrete uniform).
Often called `n` in textbooks. |

p |
A vector of success probabilities for each trial. `p` can
take on any positive value, and the vector is normalized internally. |

## Value

A `Multinomial`

object.

## Details

We recommend reading this documentation on
https://alexpghayes.github.io/distributions3, where the math
will render with additional detail and much greater clarity.

In the following, let \(X = (X_1, ..., X_k)\) be a Multinomial
random variable with success probability `p`

= \(p\). Note that
\(p\) is vector with \(k\) elements that sum to one. Assume
that we repeat the Categorical experiment `size`

= \(n\) times.

**Support**: Each \(X_i\) is in \({0, 1, 2, ..., n}\).

**Mean**: The mean of \(X_i\) is \(n p_i\).

**Variance**: The variance of \(X_i\) is \(n p_i (1 - p_i)\).
For \(i \neq j\), the covariance of \(X_i\) and \(X_j\)
is \(-n p_i p_j\).

**Probability mass function (p.m.f)**:

$$
P(X_1 = x_1, ..., X_k = x_k) = \frac{n!}{x_1! x_2! ... x_k!} p_1^{x_1} \cdot p_2^{x_2} \cdot ... \cdot p_k^{x_k}
$$

**Cumulative distribution function (c.d.f)**:

Omitted for multivariate random variables for the time being.

**Moment generating function (m.g.f)**:

$$
E(e^{tX}) = (\sum_{i=1}^k p_i e^{t_i} )^n
$$

## See also

## Examples

#> Multinomial distribution (size = 5, p = 0.3) Multinomial distribution (size = 5, p = 0.4) Multinomial distribution (size = 5, p = 0.2) Multinomial distribution (size = 5, p = 0.1)

#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#> [1,] 5 5 5 5 5 5 5 5 5 5

# pdf(X, 2)
# log_pdf(X, 2)