Stable lifecycle

dist_poisson(lambda)

Arguments

lambda

vector of (non-negative) means.

Details

Poisson distributions are frequently used to model counts.

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

Examples

dist <- dist_poisson(lambda = c(1, 4, 10)) dist
#> <distribution[3]> #> [1] Pois(1) Pois(4) Pois(10)
mean(dist)
#> [1] 1 4 10
variance(dist)
#> [1] 1 4 10
skewness(dist)
#> [1] 1.0000000 0.5000000 0.3162278
kurtosis(dist)
#> [1] 1.00 0.25 0.10
generate(dist, 10)
#> [[1]] #> [1] 1 0 0 0 1 1 1 1 1 1 #> #> [[2]] #> [1] 2 5 6 2 2 3 5 3 4 5 #> #> [[3]] #> [1] 10 5 9 10 13 14 12 8 11 13 #>
density(dist, 2)
#> [1] 0.183939721 0.146525111 0.002269996
density(dist, 2, log = TRUE)
#> [1] -1.693147 -1.920558 -6.087977
cdf(dist, 4)
#> [1] 0.99634015 0.62883694 0.02925269
quantile(dist, 0.7)
#> [1] 1 5 12