Returns the probability or pmf that a gamma-Poisson distributed random variable is equal to x. In general, the gamma Poisson functions accept arguments that are the mean parameter lambda, the overdispersion parameter sigma, and the count of interest x. When the overdispersion is equal to one, the Gamma Poisson reduces to a Poisson(lambda) distribution.

Returns the probability that a gamma-Poisson distributed random variable is less than or equal to x. In general, the gamma Poisson functions accept arguments that are the mean parameter lambda, the overdispersion parameter sigma, and the count of interest x.

Returns the smallest integer quantile for which the cumulative probability of the Gamma Poisson (lambda, sigma) distribution is larger than or equal to p.

Returns the probability that an observation from a binomial distribution with parameters p and n is less than or equal to k. In general, the binomial functions accept arguments that are the probability of success p (the event of interest), the number of trials n, and the number of successes k.

Computes the probability that a random variable from a binomial distribution is equal to k. In general, the binomial functions accept arguments that are the probability of success p (the event of interest), the number of trials n, and the number of successes k.

Returns the smallest integer quantile for which the cumulative probability of the Binomial (p, n) distribution is larger than or equal to the specified probability.

Returns the probability that a negative binomially distributed random variable is less than or equal to k, where the probability of success is p, and the number of successes is n.

Returns the probability that a negative binomially distributed random variable is equal to k, where the probability of success is p, and the number of successes is n.

Returns the probability or pmf that a beta binomially distributed random variable is less than or equal to x. In general, the beta binomial functions accept arguments that are the probability of success p (the event of interest), the overdispersion parameter delta, and the number of trials n. When the overdispersion parameter for the beta binomial is zero, the distribution reduces to a binomial(p, n).

Returns the probability or cmf that a beta binomially distributed random variable is equal to x. When the overdispersion parameter for the beta binomial is zero, the distribution reduces to a binomial(p, n).

Returns the smallest integer quantile for which the cumulative probability of the Beta Binomial (p, n, delta) distribution is larger than or equal to the specified probability. When the overdispersion parameter for the beta binomial is zero, the distribution reduces to a binomial (p, n).

Computes the probability that a random variable from a hypergeometric distribution is less than or equal to x. The hypergeometric distribution models the total number of successes in a fixed sample drawn without replacement from a finite population. The hypergeometric functions accept as arguments the size of the population N, the total number of items with the desired characteristic in the population, K, the number of samples drawn n, and the number of successes in the sample x.

Returns the smallest integer quantile for which the cumulative probability of the Poisson (lambda) distribution is larger than or equal to p.