# 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.).

#### References

 [1] A.N. Kolmogorov, "Ueber das logarithmische normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstückelung" Dokl. Akad. Nauk SSSR , 31 : 2 (1941) pp. 99–101 [2] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) [3] J. Aitchison, J.A.C. Brown, "The lognormal distribution" , Cambridge Univ. Press (1957)