Multinomial {stats} | R Documentation |

Generate multinomially distributed random number vectors and compute multinomial “density” probabilities.

rmultinom(n, size, prob) dmultinom(x, size = NULL, prob, log = FALSE)

`x` |
vector of length K of integers in `0:size` . |

`n` |
number of random vectors to draw. |

`size` |
integer, say N, specifying the total number
of objects that are put into K boxes in the typical multinomial
experiment. For `dmultinom` , it defaults to `sum(x)` . |

`prob` |
numeric non-negative vector of length K, specifying
the probability for the K classes; is internally normalized to
sum 1. |

`log` |
logical; if TRUE, log probabilities are computed. |

If `x`

is a $K$-component vector, `dmultinom(x, prob)`

is
the probability

*P(X[1]=x[1],...,X[K]=x[k]) = C * prod(j=1,..,K) p[j]^x[j]*

where *C* is the “multinomial coefficient”
*C = N! / (x[1]! * ... * 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*.

The `rmultinom()`

algorithm draws binomials from
*Bin(n[j], P[j])* sequentially, where
*n[1] = N* (N := `size`

),
*P[1] = p[1]* (*p* is `prob`

scaled to sum 1),
and for *j >= 2*, recursively
*n[j]= N - sum(k=1, .., j-1) n[k]* and
*P[j]= p[j] / (1 - sum(p[1:(j-1)]))*.

For `rmultinom()`

,
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.

`rbinom`

which is a special case conceptually.

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)

[Package *stats* version 2.2.1 Index]