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In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = \ln(X) has a normal distribution. Likewise, if Y has a normal distribution, then X = \exp(Y) has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton.^{[1]} The log-normal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.^{[1]}
A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain. The log-normal distribution is the maximum entropy probability distribution for a random variate X for which the mean and variance of \ln(X) are specified.^{[2]}
Given a log-normally distributed random variable X and two parameters \mu and \sigma that are, respectively, the mean and standard deviation of the variable’s natural logarithm, then the logarithm of X is normally distributed, and we can write X as
with Z a standard normal variable.
This relationship is true regardless of the base of the logarithmic or exponential function. If \log_a(Y) is normally distributed, then so is \log_b(Y), for any two positive numbers a,b\neq 1. Likewise, if e^X is log-normally distributed, then so is a^{X}, where a is a positive number \neq 1.
On a logarithmic scale, \mu and \sigma can be called the location parameter and the scale parameter, respectively.
In contrast, the mean, standard deviation, and variance of the non-logarithmized sample values are respectively denoted m, s.d., and v in this article. The two sets of parameters can be related as (see also Arithmetic moments below)^{[3]}
A random positive variable x is log-normally distributed if the logarithm of x is normally distributed,
A change of variables must conserve differential probability. In particular,
where
is the log-normal probability density function.^{[1]}
The cumulative distribution function is
where erfc is the complementary error function, and Φ is the cumulative distribution function of the standard normal distribution.
All moments of the log-normal distribution exist and it holds that: \operatorname{E}[X^n]=\mathrm{e}^{n\mu+\frac{n^2\sigma^2}{2}} (which can be derived by letting z=\frac{\ln(x) - (\mu+n\sigma^2)}{\sigma} within the integral). However, the expected value \operatorname{E}[e^{t X}] is not defined for any positive value of the argument t as the defining integral diverges. In consequence the moment generating function is not defined.^{[4]} The last is related to the fact that the lognormal distribution is not uniquely determined by its moments.
Similarly, the characteristic function \operatorname{E}[e^{i t X}] is not defined in the half complex plane and therefore it is not analytic in the origin. In consequence, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.^{[5]} In particular, its Taylor formal series \sum_{n=0}^\infty \frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2} diverges. However, a number of alternative divergent series representations have been obtained^{[5]}^{[6]}^{[7]}^{[8]}
A closed-form formula for the characteristic function \varphi(t) with t in the domain of convergence is not known. A relatively simple approximating formula is available in closed form and given by^{[9]}
\varphi(t)\approx\frac{\exp\bigg(-\dfrac{W^2(t\sigma^2e^\mu)+2W(t\sigma^2e^\mu)}{2\sigma^2}\bigg)}{\sqrt{1+W(t\sigma^2e^\mu)}}
where W is the Lambert W function. This approximation is derived via an asymptotic method but it stays sharp all over the domain of convergence of \varphi.
The location and scale parameters of a log-normal distribution, i.e. \mu and \sigma, are more readily treated using the geometric mean, \mathrm{GM}[X], and the geometric standard deviation, \mathrm{GSD}[X], rather than the arithmetic mean, \mathrm{E}[X], and the arithmetic standard deviation, \mathrm{SD}[X].
The geometric mean of the log-normal distribution is \mathrm{GM}[X] = e^{\mu}, and the geometric standard deviation is \mathrm{GSD}[X] = e^{\sigma}.^{[10]}^{[11]} By analogy with the arithmetic statistics, one can define a geometric variance, \mathrm{GVar}[X] = e^{\sigma^2}, and a geometric coefficient of variation,^{[10]} \mathrm{GCV}[X] = e^{\sigma} - 1.
Because the log-transformed variable Y = \ln X is symmetric and quantiles are preserved under monotonic transformations, the geometric mean of a log-normal distribution is equal to its median, \mathrm{Med}[X].^{[12]}
Note that the geometric mean is less than the arithmetic mean. This is due to the AM–GM inequality, and corresponds to the logarithm being convex down. In fact,
In finance the term e^{-\frac12\sigma^2} is sometimes interpreted as a convexity correction. From the point of view of stochastic calculus, this is the same correction term as in Itō's lemma for geometric Brownian motion.
The arithmetic mean, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are given by
respectively.
The location (\mu) and scale (\sigma) parameters can be obtained if the arithmetic mean and the arithmetic variance are known; it is simpler if \sigma is computed first:
For any real or complex number s, the s^{th} moment of a log-normally distributed variable X is given by^{[1]}
A log-normal distribution is not uniquely determined by its moments \operatorname{E}[X^k] for k\geq1, that is, there exists some other distribution with the same moments for all k.^{[1]} In fact, there is a whole family of distributions with the same moments as the log-normal distribution.
The mode is the point of global maximum of the probability density function. In particular, it solves the equation (\ln f)'=0:
The median is such a point where F_X=0.5:
The arithmetic coefficient of variation \mathrm{CV}[X] is the ratio \frac{\mathrm{SD}[X]}{\mathrm{E}[X]} (on the natural scale). For a log-normal distribution it is equal to
Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.
The partial expectation of a random variable X with respect to a threshold k is defined as g(k) = \int_k^\infty \!x{\ln\mathcal{N}}(x)\, dx where {\ln\mathcal{N}}(x) is the probability density function of X. Alternatively, and using the definition of conditional expectation, it can be written as g(k)=\operatorname{E}[X|X>k] P(X>k). For a log-normal random variable the partial expectation is given by:
Where Phi is the normal cumulative distribution function. The derivation of the formula is provided in the discussion of this WorldHeritage entry. The partial expectation formula has applications in insurance and economics, it is used in solving the partial differential equation leading to the Black–Scholes formula.
The conditional expectation of a lognormal random variable X with respect to a threshold k is its partial expectation divided by the cumulative probability of being in that range:
A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).^{[13]}
The harmonic H, geometric G and arithmetic A means of this distribution are related;^{[14]} such relation is given by
Log-normal distributions are infinitely divisible,^{[15]} but they are not stable distributions, which can be easily drawn from.^{[16]}
The log-normal distribution is important in the description of natural phenomena. The reason is that for many natural processes of growth, relative growth rate is independent of size. This is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies. It can be shown that a growth process following Gibrat's law will result in entity sizes with a log-normal distribution.^{[17]} Examples include:
For determining the maximum likelihood estimators of the log-normal distribution parameters μ and σ, we can use the same procedure as for the normal distribution. To avoid repetition, we observe that
where by L we denote the probability density function of the log-normal distribution and by \mathcal{N} that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:
Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, \ell_L and \ell_N, reach their maximum with the same \mu and \sigma. Hence, using the formulas for the normal distribution maximum likelihood parameter estimators and the equality above, we deduce that for the log-normal distribution it holds that
If \boldsymbol X \sim \mathcal{N}(\boldsymbol\mu,\,\boldsymbol\Sigma) is a multivariate normal distribution then \boldsymbol Y=\exp(\boldsymbol X) has a multivariate log-normal distribution^{[33]} with mean
and covariance matrix
In the case that all X_j have the same variance parameter \sigma_j=\sigma, these formulas simplify to
A substitute for the log-normal whose integral can be expressed in terms of more elementary functions^{[36]} can be obtained based on the logistic distribution to get an approximation for the CDF
This is a log-logistic distribution.
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