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.