Create a LogNormal distribution

A random variable created by exponentiating a Normal() distribution. Taking the log of LogNormal data returns in Normal() data.

Usage

LogNormal(log_mu = 0, log_sigma = 1)

Arguments

log_mu

The location parameter, written \(\mu\) in textbooks. Can be any real number. Defaults to 0.

log_sigma

The scale parameter, written \(\sigma\) in textbooks. Can be any positive real number. Defaults to 1.

Value

A LogNormal object.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let \(X\) be a LogNormal random variable with success probability p = \(p\).

Support: \(R^+\)

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

Variance: \([\exp(\sigma^2)-1]\exp(2\mu+\sigma^2)\)

Probability density function (p.d.f):

$$ f(x) = \frac{1}{x \sigma \sqrt{2 \pi}} \exp \left(-\frac{(\log x - \mu)^2}{2 \sigma^2} \right) $$

Cumulative distribution function (c.d.f):

$$F(x) = \frac{1}{2} + \frac{1}{2\sqrt{pi}}\int_{-x}^x e^{-t^2} dt$$

Moment generating function (m.g.f): Undefined.

See also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- LogNormal(0.3, 2)
X
#> [1] "LogNormal distribution (log_mu = 0.3, log_sigma = 2)"

random(X, 10)
#>  [1] 61.21089083 13.32648994  0.29256703  0.07317767  0.15153514  2.43630473
#>  [7]  1.36857751 13.66478070 96.47421603  2.17208867

pdf(X, 2)
#> [1] 0.09782712
log_pdf(X, 2)
#> [1] -2.324553

cdf(X, 4)
#> [1] 0.7064858
quantile(X, 0.7)
#> [1] 3.852803