| Multinom {stats} | R Documentation |
The Multinomial Distribution
Description
Generate multinomially distributed random number vectors and compute multinomial probabilities.
Usage
rmultinom(n, size, prob)
dmultinom(x, size = NULL, prob, log = FALSE)
Arguments
x |
vector of length |
n |
number of random vectors to draw. |
size |
integer, say |
prob |
numeric non-negative vector of length |
log |
logical; if TRUE, log probabilities are computed. |
Details
If x is a K-component vector, dmultinom(x, prob)
is the probability
P(X_1=x_1,\ldots,X_K=x_k) = C \times \prod_{j=1}^K
\pi_j^{x_j}
where C is the ‘multinomial coefficient’
C = N! / (x_1! \cdots x_K!)
and N = \sum_{j=1}^K x_j.
By definition, each component X_j is binomially distributed as
Bin(size, prob[j]) for j = 1, \ldots, K.
The rmultinom() algorithm draws binomials X_j from
Bin(n_j,P_j) sequentially, where
n_1 = N (N := size),
P_1 = \pi_1 (\pi is prob scaled to sum 1),
and for j \ge 2, recursively,
n_j = N - \sum_{k=1}^{j-1} X_k
and
P_j = \pi_j / (1 - \sum_{k=1}^{j-1} \pi_k).
Value
For rmultinom(),
an integer K \times n matrix where each column is a
random vector generated according to the desired multinomial law, and
hence summing to size. Whereas the transposed result
would seem more natural at first, the returned matrix is more
efficient because of columnwise storage.
Note
dmultinom is currently not vectorized at all and has
no C interface (API); this may be amended in the future.
See Also
Distributions for standard distributions, including
dbinom which is a special case conceptually.
Examples
rmultinom(10, size = 12, prob = c(0.1,0.2,0.8))
pr <- c(1,3,6,10) # normalization not necessary for generation
rmultinom(10, 20, prob = pr)
## all possible outcomes of Multinom(N = 3, K = 3)
X <- t(as.matrix(expand.grid(0:3, 0:3))); X <- X[, colSums(X) <= 3]
X <- rbind(X, 3:3 - colSums(X)); dimnames(X) <- list(letters[1:3], NULL)
X
round(apply(X, 2, function(x) dmultinom(x, prob = c(1,2,5))), 3)