distributional (version 0.6.0)

dist_logarithmic: The Logarithmic distribution

Description

The Logarithmic distribution is a discrete probability distribution derived from the logarithmic series. It is useful in modeling the abundance of species and other phenomena where the frequency of an event follows a logarithmic pattern.

Usage

dist_logarithmic(prob)

Arguments

prob

parameter. 0 <= prob < 1.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_logarithmic.html

In the following, let \(X\) be a Logarithmic random variable with parameter prob = \(p\).

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

Mean: \(\frac{-1}{\log(1-p)} \cdot \frac{p}{1-p}\)

Variance: \(\frac{-(p^2 + p\log(1-p))}{[(1-p)\log(1-p)]^2}\)

Probability mass function (p.m.f):

$$ P(X = k) = \frac{-1}{\log(1-p)} \cdot \frac{p^k}{k} $$

for \(k = 1, 2, 3, \ldots\)

Cumulative distribution function (c.d.f):

The c.d.f. does not have a simple closed form. It is computed using the recurrence relationship \(P(X = k+1) = \frac{p \cdot k}{k+1} \cdot P(X = k)\) starting from \(P(X = 1) = \frac{-p}{\log(1-p)}\).

Moment generating function (m.g.f):

$$ E(e^{tX}) = \frac{\log(1 - pe^t)}{\log(1-p)} $$

for \(pe^t < 1\)

See Also

actuar::Logarithmic

Examples

dist <- dist_logarithmic(prob = c(0.33, 0.66, 0.99))
dist

if (FALSE) { # requireNamespace("actuar", quietly = TRUE)
mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)
}