distributional (version 0.5.0)

dist_poisson: The Poisson Distribution

Description

Poisson distributions are frequently used to model counts.

Usage

dist_poisson(lambda)

Arguments

lambda

vector of (non-negative) means.

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

In the following, let \(X\) be a Poisson random variable with parameter lambda = \(\lambda\).

Support: \(\{0, 1, 2, 3, ...\}\)

Mean: \(\lambda\)

Variance: \(\lambda\)

Probability mass function (p.m.f):

$$ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} $$

Cumulative distribution function (c.d.f):

$$ P(X \le k) = e^{-\lambda} \sum_{i = 0}^{\lfloor k \rfloor} \frac{\lambda^i}{i!} $$

Moment generating function (m.g.f):

$$ E(e^{tX}) = e^{\lambda (e^t - 1)} $$

See Also

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)