A lognormal distribution is the distribution of a random variable whose logarithm has a normal distribution.

Overview

A lognormal distribution is the distribution of a random variable whose logarithm has a normal distribution. It therefore can be seen to arise as the result of many small independent multiplicative effects. All moments exist, but this distribution does not have a moment generating function and is not completely determined by its moments. Like the Gamma distribution, it has variance (roughly) proportional to the square of the mean, but the lognormal has a heavier right tail. Datasets thought to be lognormal are often analyzed on a log scale (the logarithms will be normal).

Univariate distribution

The univariate lognormal takes values on the positive half-line and is right-skew. Its density function can be obtained by making the substitution $\log{x}$ for $x$ and applying the change of variables rule to the density function of the standard normal distribution, $\mathcal{N}(x;\mu,\sigma^2)$. Explicitly, the density function is:

$$\frac{1}{x\sigma\sqrt{2\pi}} \exp \left( - \frac{(\log x - \mu)^2}{2\sigma^2}\right) $$

Multivariate distribution

The univariate distribution can be generalized to a $n$-dimensional multivariate distribution with mean vector $\mu$ and covariance matrix $\Sigma$. The density function in this case is:

$$\frac{1}{(2\pi)^{\frac{n}{2}} |\Sigma|^{\frac{1}{2}} \prod_{i=1}^{n}x_i } \exp \left( -\frac{1}{2} (\log x - \mu)^T \Sigma^{-1} (\log x - \mu) \right)$$

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