Poisson distributions are frequently used to model counts.

Poisson(lambda)

## 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://alexpghayes.github.io/distributions3, where the math will render with additional detail.

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

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):

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

Other discrete distributions: Bernoulli, Binomial, Categorical, Geometric, HyperGeometric, Multinomial, NegativeBinomial

## Examples


set.seed(27)

X <- Poisson(2)
X#> Poisson distribution (lambda = 2)
random(X, 10)#>   5 0 4 1 1 1 0 0 1 1
pdf(X, 2)#>  0.2706706log_pdf(X, 2)#>  -1.306853
cdf(X, 4)#>  0.947347quantile(X, 0.7)#>  3
cdf(X, quantile(X, 0.7))#>  0.8571235quantile(X, cdf(X, 7))#>  7