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#' The Poisson Distribution
#'
#' @description
#' `r lifecycle::badge('stable')`
#'
#' Poisson distributions are frequently used to model counts.
#'
#' @inheritParams stats::dpois
#'
#'
#' @details
#'
#' We recommend reading this documentation on
#' <https://pkg.mitchelloharawild.com/distributional/>, where the math
#' will render nicely.
#'
#' In the following, let \eqn{X} be a Poisson random variable with parameter
#' `lambda` = \eqn{\lambda}.
#'
#' **Support**: \eqn{\{0, 1, 2, 3, ...\}}{{0, 1, 2, 3, ...}}
#'
#' **Mean**: \eqn{\lambda}
#'
#' **Variance**: \eqn{\lambda}
#'
#' **Probability mass function (p.m.f)**:
#'
#' \deqn{
#' P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
#' }{
#' P(X = k) = \lambda^k e^(-\lambda) / k!
#' }
#'
#' **Cumulative distribution function (c.d.f)**:
#'
#' \deqn{
#' P(X \le k) = e^{-\lambda}
#' \sum_{i = 0}^{\lfloor k \rfloor} \frac{\lambda^i}{i!}
#' }{
#' P(X \le k) = e^(-\lambda)
#' \sum_{i = 0}^k \lambda^i / i!
#' }
#'
#' **Moment generating function (m.g.f)**:
#'
#' \deqn{
#' E(e^{tX}) = e^{\lambda (e^t - 1)}
#' }{
#' E(e^(tX)) = e^(\lambda (e^t - 1))
#' }
#' @seealso [stats::Poisson]
#'
#' @examples
#' dist <- dist_poisson(lambda = c(1, 4, 10))
#'
#' dist
#' mean(dist)
#' variance(dist)
#' skewness(dist)
#' kurtosis(dist)
#'
#' generate(dist, 10)
#'
#' density(dist, 2)
#' density(dist, 2, log = TRUE)
#'
#' cdf(dist, 4)
#'
#' quantile(dist, 0.7)
#'
#' @name dist_poisson
#' @export
dist_poisson <- function(lambda){
lambda <- vec_cast(lambda, double())
if(any(lambda < 0)){
abort("The lambda parameter of an Poisson distribution must be non-negative.")
}
new_dist(l = lambda, class = "dist_poisson")
}
#' @export
format.dist_poisson <- function(x, digits = 2, ...){
sprintf(
"Pois(%s)",
format(x[["l"]], digits = digits, ...)
)
}
#' @export
density.dist_poisson <- function(x, at, ...){
stats::dpois(at, x[["l"]])
}
#' @export
log_density.dist_poisson <- function(x, at, ...){
stats::dpois(at, x[["l"]], log = TRUE)
}
#' @export
quantile.dist_poisson <- function(x, p, ...){
stats::qpois(p, x[["l"]])
}
#' @export
cdf.dist_poisson <- function(x, q, ...){
stats::ppois(q, x[["l"]])
}
#' @export
generate.dist_poisson <- function(x, times, ...){
as.integer(stats::rpois(times, x[["l"]]))
}
#' @export
mean.dist_poisson <- function(x, ...){
x[["l"]]
}
#' @export
covariance.dist_poisson <- function(x, ...){
x[["l"]]
}
#' @export
skewness.dist_poisson <- function(x, ...) 1 / sqrt(x[["l"]])
#' @export
kurtosis.dist_poisson <- function(x, ...) 1 / x[["l"]]
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