Multinomial {stats} | R Documentation

## The Multinomial Distribution

### Description

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

### Usage

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

### Arguments

`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.

### Details

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)])).

### Value

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.

### Note

`dmultinom` is currently not vectorized at all and has no C interface (API); this may be amended in the future.

### See Also

`rbinom` 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)

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