Generate multinomially distributed random number vectors and compute multinomial probabilities.
vector of length K of integers in
number of random vectors to draw.
integer, say N, specifying the total number
of objects that are put into K boxes in the typical multinomial
numeric non-negative vector of length K, specifying the probability for the K classes; is internally normalized to sum 1. Infinite and missing values are not allowed.
logical; if TRUE, log probabilities are computed.
x is a K-component vector,
is the probability
P(X=x, … , X[K]=x[k]) = C * prod(j=1 , …, K) p[j]^x[j]
where C is the ‘multinomial coefficient’
C = N! / (x! * … * 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, …, K.
rmultinom() algorithm draws binomials X[j] from
Bin(n[j], P[j]) sequentially, where
n = N (N :=
P = p (p is
prob scaled to sum 1),
and for j ≥ 2, recursively,
n[j] = N - sum(k=1, …, j-1) X[k]
P[j] = p[j] / (1 - sum(p[1:(j-1)])).
an integer K x 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.
dmultinom is currently not vectorized at all and has
no C interface (API); this may be amended in the future.
Distributions for standard distributions, including
dbinom which is a special case conceptually.
1 2 3 4 5 6 7 8 9 10
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)
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