The Logarithmic distribution

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.

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 $pe^t < 1$

Examples

dist <- dist_logarithmic(prob = c(0.33, 0.66, 0.99))
dist
#> <distribution[3]>
#> [1] Logarithmic(0.33) Logarithmic(0.66) Logarithmic(0.99)

mean(dist)
#> [1]  1.229875  1.799369 21.497577
variance(dist)
#> [1]    0.3230419    2.0545331 1687.6118748
support(dist)
#> <support_region[3]>
#> [1] N+ N+ N+
generate(dist, 10)
#> [[1]]
#>  [1] 1 1 1 2 1 1 1 1 1 1
#> 
#> [[2]]
#>  [1] 1 2 1 1 9 1 1 3 1 3
#> 
#> [[3]]
#>  [1]  20   2   1   7 105   1   6   2  23   1
#> 

density(dist, 2)
#> [1] 0.1359627 0.2018892 0.1064130
density(dist, 2, log = TRUE)
#> [1] -1.995375 -1.600036 -2.240427

cdf(dist, 4)
#> [1] 0.9972938 0.9464773 0.4437691

quantile(dist, 0.7)
#> [1]  1  2 16

Arguments

Details

See also

Examples