Create a Poisson distribution

Poisson distributions are frequently used to model counts.

Usage

Poisson(lambda = numeric())

Arguments

lambda

The shape parameter, which is also the mean and the variance of the distribution. Can be any positive number.

Value

A Poisson object.

Details

We recommend reading this documentation on https://zeileis.github.io/distributions3/, where the math will render with additional detail.

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

Other discrete distributions: Bernoulli(), Binomial(), Categorical(), Geometric(), HurdleNegativeBinomial(), HurdlePoisson(), HyperGeometric(), Multinomial(), NegativeBinomial(), PoissonBinomial(), ZINegativeBinomial(), ZIPoisson(), ZTNegativeBinomial(), ZTPoisson()

Examples

set.seed(27)

X <- Poisson(2)
X
#> [1] "Poisson(lambda = 2)"

random(X, 10)
#>  [1] 5 0 4 1 1 1 0 0 1 1

pdf(X, 2)
#> [1] 0.2706706
log_pdf(X, 2)
#> [1] -1.306853

cdf(X, 4)
#> [1] 0.947347
quantile(X, 0.7)
#> [1] 3

cdf(X, quantile(X, 0.7))
#> [1] 0.8571235
quantile(X, cdf(X, 7))
#> [1] 7