Returns the cumulative distribution function (cdf) of the beta binomial distribution. This is the probability that a beta binomially distributed random variable is less than or equal to k. The cdf is calculated as the summation of the beta binomial pmf for values of X from 0 to k.
The count of interest; k must be an integer.
The overdispersion parameter, which must be between Maximum[-p/(n-p-1), -(1-p)/(n-2+p)] and 1. When the overdispersion parameter is zero, the distribution reduces to Binomial(n, p).
Returns the probability mass function (pmf) of the beta binomial distribution. This is the probability that a beta binomially distributed random variable is equal to k. The pmf is parameterized as follows:
The count of interest; k must be an integer.
The overdispersion parameter, which must be between Maximum[-p/(n-p-1), -(1-p)/(n-2+p)] and 1. When the overdispersion parameter is zero, the distribution reduces to Binomial(n, p).
The beta binomial distribution results from assuming that X|π follows a Binomial(n,π) distribution and π follows a Beta(p(1-δ)/δ,(1-p)(1-δ)/δ) distribution. It is useful when the data are a combination of several Binomial distributions that each have different probabilities of success. For more information, see Distributions topic in the Basic Analysis book.
The overdispersion parameter, which must be between Maximum[-p/(n-p-1), -(1-p)/(n-2+p)] and 1. When the overdispersion parameter is zero, the distribution reduces to Binomial(n, p).
The cumulative probability of the quantile desired; cumprob must be between 0 and 1.
Returns the cumulative distribution function (cdf) of the binomial distribution. This is the probability that a binomially distributed random variable is less than or equal to k. The cdf is calculated as the summation of the binomial pmf for values of X from 0 to k.
Returns the probability mass function (pmf) of the binomial distribution. This is the probability that a binomially distributed variable is equal to k. The pmf is parameterized as follows:
The cumulative probability of the quantile desired; cumprob must be between 0 and 1.
Returns the cumulative distribution function (cdf) of the gamma-Poisson distribution. This is the probability that a gamma-Poisson distributed random variable is less than or equal to k. The cdf is calculated as the summation of the gamma-Poisson pmf for values of X from 0 to k.
The count of interest; k must be an integer.
The shape parameter, λ, which much be greater than 0. This is the mean of the distribution.
The overdispersion parameter, σ, which must be greater than or equal to 1. When the overdispersion parameter is 1, the distribution reduces to a Poisson(λ) distribution.
Returns the probability mass function (pmf) of the gamma-Poisson distribution. This is the probability that a gamma-Poisson distributed random variable is equal to k. The pmf is parameterized as follows:
is the Gamma function.The count of interest; k must be an integer.
The shape parameter λ, which much be greater than 0. This is the mean of the distribution.
The overdispersion parameter σ, which must be greater than or equal to 1. When the overdispersion parameter is 1, the distribution reduces to a Poisson(λ) distribution.
The gamma Poisson distribution results from assuming that X|μ follows a Poisson(μ) distribution and μ follows a Gamma(λ/(σ-1),σ-1) distribution. It is useful when the data are a combination of several Poisson(μ) distributions that each have different values of μ. For more information, see Distributions topic in the Basic Analysis book.
The shape parameter λ, which much be greater than 0. This is the mean of the distribution.
The overdispersion parameter σ, which must be greater than or equal to 1. When the overdispersion parameter is 1, the distribution reduces to a Poisson(λ) distribution.
The cumulative probability of the quantile desired; cumprob must be between 0 and 1.
Returns the cumulative distribution function (cdf) of the hypergeometric distribution. This is the probability that a hypergeometrically distributed random variable is less than or equal to x. The cdf is calculated as the summation of the hypergeometric pmf for values of X from 0 to x.
Returns the probability mass function (pmf) of the hypergeometric distribution. This is the probability that a hypergeometrically distributed random variable is equal to x. The pmf is parameterized as follows:
Returns the cumulative distribution function (cdf) of the negative binomial distribution. This is the probability that a negative binomially distributed random variable is less than or equal to k. The cdf is calculated as the summation of the negative binomial pmf for values of X from 0 to k.
Returns the probability mass function (pmf) of the negative binomial distribution. This is the probability that a negative binomially distributed random variable is equal to k. The pmf is parameterized as follows:
The return value of the pmf is the probability of observing the nth success after k failures have occurred.
Returns the cumulative distribution function (cdf) of the Poisson distribution. This is the probability that a Poisson distributed random variable with mean lambda is less than or equal to k. The cdf is calculated as the summation of the Poisson pmf for values of X from 0 to k.
The number of events in a given time interval; k must be an integer.
The shape parameter λ, which much be greater than 0. This is the mean of the distribution.
Returns the probability mass function (pmf) of the Poisson distribution. This is the probability that a Poisson distributed random variable with mean lambda is equal to k. The pmf is parameterized as follows:
The number of events in a given time interval; k must be an integer.
The cumulative probability of the quantile desired; cumprob must be between 0 and 1.