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

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):

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

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

Examples

set.seed(27) X <- LogNormal(0.3, 2) X
#> 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