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] 2.268372 2.479770 2.808937 2.772599 2.424468 2.700803 2.531545 2.237002
#>  [9] 2.802470 2.337099
#> 
#> [[2]]
#>  [1] 7.036684 8.971989 7.454462 7.048715 7.657213 9.429064 8.038234 8.035449
#>  [9] 7.258841 7.143602
#> 
#> [[3]]
#>  [1] 24.24908 19.85483 18.04264 20.43742 21.33054 20.28091 19.40403 20.84049
#>  [9] 17.22014 15.53991
#> 
#> [[4]]
#>  [1] 57.09642 52.17184 54.26868 51.47233 55.98393 55.31401 55.75467 48.11861
#>  [9] 51.09322 52.22699
#> 
#> [[5]]
#>  [1] 139.9108 141.0417 143.7527 163.0416 156.7406 162.1838 140.0949 139.0415
#>  [9] 157.3752 126.5862
#> 

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)