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):

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

## Examples

set.seed(27) X <- Poisson(2) X
#> Poisson distribution (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