The log-normal distribution

The log-normal distribution is a commonly used transformation of the Normal distribution. If \(X\) follows a log-normal distribution, then \(\ln{X}\) would be characterised by a Normal distribution.

dist_lognormal(mu = 0, sigma = 1)

Arguments

mu

The mean (location parameter) of the distribution, which is the mean of the associated Normal distribution. Can be any real number.

sigma

The standard deviation (scale parameter) of the distribution. Can be any positive number.

Details

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

In the following, let \(X\) be a log-normal random variable with mu = \(\mu\) and sigma = \(\sigma\).

Support: \(R^+\), the set of positive real numbers.

Mean: \(e^{\mu + \sigma^2/2}\)

Variance: \((e^{\sigma^2} - 1) e^{2\mu + \sigma^2}\)

Skewness: \((e^{\sigma^2} + 2) \sqrt{e^{\sigma^2} - 1}\)

Excess Kurtosis: \(e^{4\sigma^2} + 2 e^{3\sigma^2} + 3 e^{2\sigma^2} - 6\)

Probability density function (p.d.f):

$$ f(x) = \frac{1}{x\sqrt{2 \pi \sigma^2}} e^{-(\ln{x} - \mu)^2 / (2 \sigma^2)} $$

Cumulative distribution function (c.d.f):

$$ F(x) = \Phi\left(\frac{\ln{x} - \mu}{\sigma}\right) $$

where \(\Phi\) is the c.d.f. of the standard Normal distribution.

Moment generating function (m.g.f):

Does not exist in closed form.

See also

stats::Lognormal

Examples

dist <- dist_lognormal(mu = 1:5, sigma = 0.1)

dist
#> <distribution[5]>
#> [1] lN(1, 0.01) lN(2, 0.01) lN(3, 0.01) lN(4, 0.01) lN(5, 0.01)
mean(dist)
#> [1]   2.731907   7.426094  20.186216  54.871824 149.157083
variance(dist)
#> [1]   0.07500759   0.55423526   4.09527545  30.26022006 223.59446360
skewness(dist)
#> [1] 0.3017591 0.3017591 0.3017591 0.3017591 0.3017591
kurtosis(dist)
#> [1] 0.1623239 0.1623239 0.1623239 0.1623239 0.1623239

generate(dist, 10)
#> [[1]]
#>  [1] 2.237002 2.802470 2.337099 2.588651 3.300610 2.742343 2.593077 2.816931
#>  [9] 3.468759 2.957101
#> 
#> [[2]]
#>  [1] 8.035449 7.258841 7.143602 8.920738 7.304185 6.637518 7.518505 7.847068
#>  [9] 7.460930 7.138343
#> 
#> [[3]]
#>  [1] 20.84049 17.22014 15.53991 21.00460 19.19295 19.96433 18.93561 20.59534
#>  [9] 20.34889 20.51100
#> 
#> [[4]]
#>  [1] 48.11861 51.09322 52.22699 51.47032 51.88633 52.88368 59.97967 57.66163
#>  [9] 59.66409 51.53802
#> 
#> [[5]]
#>  [1] 139.0415 157.3752 126.5862 157.6473 160.2252 160.7463 147.0482 135.2018
#>  [9] 152.9690 173.9745
#> 

density(dist, 2)
#> [1]  1.799910e-02  1.637111e-37 5.539330e-116 6.972494e-238  0.000000e+00
density(dist, 2, log = TRUE)
#> [1]   -4.017433  -84.702715 -265.387997 -546.073279 -926.758561

cdf(dist, 4)
#> [1]  9.999440e-01  4.203228e-10  7.003186e-59 6.915322e-151 2.970982e-286

quantile(dist, 0.7)
#> [1]   2.864632   7.786878  21.166930  57.537681 156.403632

# A log-normal distribution X is exp(Y), where Y is a Normal distribution of
# the same parameters. So log(X) will produce the Normal distribution Y.
log(dist)
#> <distribution[5]>
#> [1] N(1, 0.01) N(2, 0.01) N(3, 0.01) N(4, 0.01) N(5, 0.01)