Editing Log-normal distribution (section)

=== Probability density function ===

A positive random variable <math> X </math> is log-normally distributed (i.e., {{nowrap|<math display="inline"> X \sim \operatorname{Lognormal} \left( \mu, \sigma^2 \right) </math>),}} if the natural logarithm of <math> X </math> is normally distributed with mean <math> \mu</math> and variance {{nowrap|<math> \sigma^2</math>:}}

<math display="block"> \ln X \sim \mathcal{N}(\mu,\sigma^2)</math>

Let <math> \Phi </math> and <math> \varphi </math> be respectively the cumulative probability distribution function and the probability density function of the <math> \mathcal{N}( 0, 1 ) </math> standard normal distribution, then we have that<ref name=":1"/><ref name="JKB"/> the [[probability density function]] of the log-normal distribution is given by:

<math display="block">\begin{align}
f_X(x) & = \frac{d}{dx} \Pr\nolimits_X\left[ X \le x \right] \\[6pt]
& = \frac{d}{dx} \Pr\nolimits_X\left[ \ln X \le \ln x \right] \\[6pt]
& = \frac{d}{dx} \Phi{\left( \frac{ \ln x -\mu }{ \sigma } \right)} \\[6pt]
& = \varphi{\left( \frac{\ln x - \mu} \sigma \right)} \frac{d}{dx} \left( \frac{ \ln x - \mu }{ \sigma }\right) \\[6pt]
& = \varphi{\left( \frac{ \ln x - \mu }{ \sigma } \right)} \frac{ 1 }{ \sigma x } \\[6pt]
& = \frac{ 1 }{ x \sigma\sqrt{2 \pi } } \exp\left( -\frac{ (\ln x-\mu)^2 }{2 \sigma^2} \right) ~.
\end{align}</math>