Description Usage Arguments Details Value See Also Examples

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.

1 | ```
Multinomial(size, p)
``` |

`size` |
The number of trials. Must be an integer greater than or equal
to one. When |

`p` |
A vector of success probabilities for each trial. |

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) = n! / (x_1! x_2! ... x_k!) p_1^x_1 p_2^x_2 ... 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)) = (p_1 e^t_1 + p_2 e^t_2 + ... + p_k e^t_k)^n
*

A `Multinomial`

object.

Other discrete distributions: `Bernoulli`

,
`Binomial`

, `Categorical`

,
`Geometric`

, `HyperGeometric`

,
`NegativeBinomial`

, `Poisson`

1 2 3 4 5 6 7 8 9 | ```
set.seed(27)
X <- Multinomial(size = 5, p = c(0.3, 0.4, 0.2, 0.1))
X
random(X, 10)
# pdf(X, 2)
# log_pdf(X, 2)
``` |

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