`pmultinom`

is a library for calculating multinomial
probabilities. The probabilities that can be calculated include the
multinomial cumulative distribution function: \[P(N_1 \le u_1, N_2 \le u_2, \cdots, N_k \le
u_k)\] In this case the usage would be

`pmultinom(upper=us, size=n, probs=ps, method="exact")`

where `us`

is the vector containing \(u_1, u_2, \cdots, u_k\), and `n`

and `ps`

are the parameters of the multinomial distribution.
This usage is analogous to the use of `pbinom`

. Another
important case is the probability of seeing more than some minimum
number of observations in each category: \[P(N_1 > l_1, N_2 > l_2, \cdots, N_k >
l_k)\] In this case the usage would be

`pmultinom(lower=ls, size=n, probs=ps, method="exact")`

where this time `ls`

is the vector containing \(l_1, l_2, \cdots, l_k\). Notice that in
this case these are greater than signs, not greater than or equal signs.
This is analogous to the usage of `pbinom`

with
`lower.tail=FALSE`

. With some creativity, these can be
adapted to calculate the probability that the maximum or minimum of a
multinomial random vector is a given number, or that a given category
will be the most or least observed. `pmultinom`

also supports
a more general usage, in which both lower and upper bounds are
specified: \[P(l_1 < N_1 \le u_1, l_2 <
N_2 \le u_2, \cdots, l_k < N_k \le u_k)\] In this case the
usage would be

`pmultinom(lower=ls, upper=us, size=n, probs=ps, method="exact")`

See `vignette("pmultinom")`

for the above text in Latex
plus an example application. Many thanks to Aislyn Schalck for advice
and encouragement.