distributions3 (version 0.1.1)

LogNormal: Create a LogNormal distribution

Description

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):

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

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

Other continuous distributions: Beta, Cauchy, ChiSquare, Exponential, FisherF, Gamma, Logistic, Normal, StudentsT, Tukey, Uniform, Weibull

Examples

# NOT RUN {
set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

# }