Multinomial {distributions3} | R Documentation |
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)
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) = \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}) = \left(\sum_{i=1}^k p_i e^{t_i}\right)^n
A Multinomial
object.
Other discrete distributions:
Bernoulli()
,
Binomial()
,
Categorical()
,
Geometric()
,
HurdleNegativeBinomial()
,
HurdlePoisson()
,
HyperGeometric()
,
NegativeBinomial()
,
Poisson()
,
ZINegativeBinomial()
,
ZIPoisson()
,
ZTNegativeBinomial()
,
ZTPoisson()
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)