# Logarithmic normal distribution

The continuous probability distribution, concentrated on $( 0 , \infty )$, with density

$$\tag{* } p ( x) = \ \left \{ \begin{array}{ll} \frac{ \mathop{\rm log} e }{\sigma x \sqrt {2 \pi } } e ^ {- ( \mathop{\rm log} x - a ) ^ {2} / 2 \sigma ^ {2} } , & x > 0 , \\ 0 , & x \leq 0 , \\ \end{array} \right .$$

where $- \infty < a < \infty$, $\sigma ^ {2} > 0$. A random variable $X$ is subject to the logarithmic normal distribution with density (*) if its logarithm $\mathop{\rm log} X$ has the normal distribution with parameters $a$ and $\sigma ^ {2}$. Thus, $a = {\mathsf E} \mathop{\rm log} X$ and $\sigma ^ {2} = {\mathsf D} \mathop{\rm log} X$. The logarithmic normal distribution is a unimodal distribution and has positive asymmetry. The moments of a random variable $X$ with logarithmic normal distribution with parameters $a$ and $\sigma ^ {2}$ are given by the formula

$${\mathsf E} X ^ {k} = \ e ^ {k a + k ^ {2} \sigma ^ {2} / 2 } ;$$

hence the mean and variance are, respectively, equal to

$${\mathsf E} X = e ^ {a + \sigma ^ {2} / 2 } \ \ \textrm{ and } \ \ {\mathsf D} X = e ^ {2 a + \sigma ^ {2} } ( e ^ {\sigma ^ {2} } - 1 ) .$$

The logarithmic normal distribution is one of the simplest examples of a distribution that is not defined uniquely by its moments. The properties of the logarithmic normal distribution are determined by the properties of the corresponding normal distribution. An important property of the logarithmic normal distribution is the following: The product of independent random variables with a logarithmic normal distribution is again subject to a logarithmic normal distribution. There is an analogue of the central limit theorem: The distribution of the product of $n$ independent positive random variables tends, under certain general conditions, to a logarithmic normal distribution as $n \rightarrow \infty$. The logarithmic normal distribution arises as a limit distribution and in certain other schemes (for example, in models of branching of particles, models of growth, etc.).

How to Cite This Entry:
Logarithmic normal distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmic_normal_distribution&oldid=47702
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article