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 characteristed 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 https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

In the following, let \(Y\) be a Normal random variable with mean mu = \(\mu\) and standard deviation sigma = \(\sigma\). The log-normal distribution \(X = exp(Y)\) is characterised by:

Support: \(R+\), the set of all real numbers greater than or equal to 0.

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

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

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):

The cumulative distribution function has the form

$$ F(x) = \Phi((\ln{x} - \mu)/\sigma) $$

Where \(Phi\) is the CDF of a standard Normal distribution, N(0,1).

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] 2.588997 2.979417 2.846744 3.458272 2.911521 2.398239 2.753209 2.487294
#>  [9] 2.984255 2.985225
#> 
#> [[2]]
#>  [1] 7.760536 7.330519 7.956947 7.571848 7.344098 6.322617 7.253922 6.761270
#>  [9] 7.867682 7.990143
#> 
#> [[3]]
#>  [1] 16.94034 22.88317 18.27328 19.49396 19.86043 19.08899 18.83538 22.31345
#>  [9] 18.21033 21.28038
#> 
#> [[4]]
#>  [1] 56.11891 56.17060 50.40977 54.04689 58.78795 44.94419 52.20205 55.64627
#>  [9] 47.06076 46.84202
#> 
#> [[5]]
#>  [1] 140.4550 145.9837 137.0261 162.8232 120.1060 134.6020 148.9233 155.3421
#>  [9] 147.0364 149.3243
#> 

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)