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] 2.840998 2.547219 3.006535 2.690661 2.807004 2.611946 2.358359 2.482696
#>  [9] 2.952893 2.384222
#> 
#> [[2]]
#>  [1] 8.222861 7.479137 8.649743 8.659774 6.384512 8.145930 8.260255 9.600144
#>  [9] 6.190430 5.345526
#> 
#> [[3]]
#>  [1] 20.71173 19.73855 21.45053 20.06781 20.95095 21.43447 20.88472 18.59968
#>  [9] 21.88265 19.38821
#> 
#> [[4]]
#>  [1] 56.86405 55.60589 45.25569 51.87961 53.74228 61.65259 54.82914 56.81161
#>  [9] 62.11890 59.54112
#> 
#> [[5]]
#>  [1] 157.8453 127.9634 150.4482 168.9324 130.4428 134.5977 173.8252 145.4930
#>  [9] 151.0819 163.0845
#> 

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)