Editing Log-normal distribution (section)

===Generation and parameters===

Let <math> Z </math> be a [[Normal distribution#Standard normal distribution|standard normal variable]], and let <math>\mu</math> and <math>\sigma</math> be two real numbers, with {{nowrap|<math>\sigma > 0</math>.}} Then, the distribution of the random variable

<math display="block"> X = e^{\mu + \sigma Z} </math>

is called the log-normal distribution with parameters <math>\mu</math> and {{nowrap|<math>\sigma</math>.}} These are the [[expected value]] (or [[mean]]) and [[standard deviation]] of the variable's natural [[logarithm]], {{nowrap|<math>\ln X </math>,}} ''not'' the expectation and standard deviation of <math> X </math> itself.
[[File:Lognormal Distribution.svg|thumb|upright=1.5|Relation between normal and log-normal distribution. If <math> Y = \mu + \sigma Z </math> is normally distributed, then <math> X \sim e^Y </math> is log-normally distributed.]]

This relationship is true regardless of the base of the logarithmic or exponential function: If <math>\log_a X </math> is normally distributed, then so is <math>\log_b X </math> for any two positive numbers {{nowrap|<math> a, b \neq 1</math>.}} Likewise, if <math> e^Y </math> is log-normally distributed, then so is {{nowrap|<math> a^Y </math>,}} where {{nowrap|<math>0 < a \neq 1</math>.}}

In order to produce a distribution with desired mean <math>\mu_X</math> and variance {{nowrap|<math> \sigma_X^2</math>,}} one uses <math> \mu = \ln \frac{ \mu_X^2 }{ \sqrt{ \mu_X^2 + \sigma_X^2 } } </math> and {{nowrap|<math> \sigma^2 = \ln\left( 1 + \frac{ \sigma_X^2 }{ \mu_X^2 } \right) </math>.}}

Alternatively, the "multiplicative" or "geometric" parameters <math> \mu^* = e^\mu </math> and <math> \sigma^* = e^\sigma </math> can be used. They have a more direct interpretation: <math> \mu^* </math> is the '''''[[median]]''''' of the distribution, and <math> \sigma^* </math> is useful for determining "scatter" intervals, see below.