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] 2.647334 2.809561 2.645544 2.570626 2.588997 2.979417 2.846744 3.458272
#>  [9] 2.911521 2.398239
#> 
#> [[2]]
#>  [1] 7.483997 6.761166 8.112045 8.114684 7.760536 7.330519 7.956947 7.571848
#>  [9] 7.344098 6.322617
#> 
#> [[3]]
#>  [1] 19.71820 18.37904 21.38658 21.71946 16.94034 22.88317 18.27328 19.49396
#>  [9] 19.86043 19.08899
#> 
#> [[4]]
#>  [1] 51.19987 60.65425 49.50082 57.84606 56.11891 56.17060 50.40977 54.04689
#>  [9] 58.78795 44.94419
#> 
#> [[5]]
#>  [1] 141.8999 151.2622 127.9244 127.3298 140.4550 145.9837 137.0261 162.8232
#>  [9] 120.1060 134.6020
#> 

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)