From Wikipedia, the free encyclopedia  View original article
Probability density function Some lognormal density functions with identical location parameter μ but differing scale parameters σ  
Cumulative distribution function Cumulative distribution function of the lognormal distribution (with μ = 0 )  
Notation  

Parameters  σ^{2} > 0 — shape (real), μ ∈ R — logscale 
Support  x ∈ (0, +∞) 
CDF  
Mean  
Median  
Mode  
Variance  
Skewness  
Ex. kurtosis  
Entropy  
MGF  (defined only on the negative halfaxis, see text) 
CF  representation is asymptotically divergent but sufficient for numerical purposes 
Fisher information 
Probability density function Some lognormal density functions with identical location parameter μ but differing scale parameters σ  
Cumulative distribution function Cumulative distribution function of the lognormal distribution (with μ = 0 )  
Notation  

Parameters  σ^{2} > 0 — shape (real), μ ∈ R — logscale 
Support  x ∈ (0, +∞) 
CDF  
Mean  
Median  
Mode  
Variance  
Skewness  
Ex. kurtosis  
Entropy  
MGF  (defined only on the negative halfaxis, see text) 
CF  representation is asymptotically divergent but sufficient for numerical purposes 
Fisher information 
In probability theory, a lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is lognormally distributed, then has a normal distribution. Likewise, if has a normal distribution, then has a lognormal distribution. A random variable which is lognormally distributed takes only positive real values.
Lognormal is also written log normal or lognormal. The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton.^{[1]} The lognormal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.^{[1]}
A variable might be modeled as lognormal if it can be thought of as the multiplicative product of many independent random variables each of which is positive. (This is justified by considering the central limit theorem in the logdomain.) For example, in finance, the variable could represent the compound return from a sequence of many trades (each expressed as its return + 1); or a longterm discount factor can be derived from the product of shortterm discount factors. In wireless communication, the sas caused by shadowing or slow fading from random objects is often assumed to be lognormally distributed: see logdistance path loss model.
The lognormal distribution is the maximum entropy probability distribution for a random variate X for which the mean and variance of are fixed.^{[2]}
In a lognormal distribution X, the parameters denoted μ and σ are, respectively, the mean and standard deviation of the variable’s natural logarithm (by definition, the variable’s logarithm is normally distributed), which means
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 ≠ 1. Likewise, if is lognormally distributed, then so is , where is a positive number ≠ 1.
On a logarithmic scale, μ and σ can be called the location parameter and the scale parameter, respectively.
In contrast, the mean and standard deviation of the nonlogarithmized sample values are denoted m and s.d. in this article.
A lognormal distribution with mean m and variance v has parameters^{[3]}
The probability density function of a lognormal distribution is:^{[1]}
This follows by applying the changeofvariables rule on the density function of a normal distribution.
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 lognormal distributions exist and it holds that
However, the moment generating function
does not converge.
The characteristic function, E[e^{ itX}], has a number of representations.^{[citation needed]} The integral itself converges for Im(t) ≤ 0. The simplest representation is obtained by Taylor expanding e^{ itX} and using formula for moments below, giving^{[citation needed]}
This series representation is divergent for Re(σ^{2}) > 0.^{[citation needed]} However, it is sufficient for evaluating the characteristic function numerically at positive as long as the upper limit in the sum above is kept bounded, n ≤ N, where
and σ^{2} < 0.1.^{[citation needed]} To bring the numerical values of parameters μ, σ into the domain where strong inequality holds true one could use the fact that if X is lognormally distributed then X^{m} is also lognormally distributed with parameters μm, σm. Since , the inequality could be satisfied for sufficiently small m. The sum of series first converges to the value of φ(t) with arbitrary high accuracy if m is small enough, and left part of the strong inequality is satisfied. If considerably larger number of terms are taken into account the sum eventually diverges when the right part of the strong inequality is no longer valid.
Another useful representation is available^{[4]}^{[5]} by means of double Taylor expansion of e^{(ln x − μ)2/(2σ2)}.
The momentgenerating function for the lognormal distribution does not exist on the domain R, but only exists on the halfinterval (−∞, 0].^{[citation needed]}
For the lognormal distribution, the location and scale properties of the distribution are more readily treated using the geometric mean and geometric standard deviation than the arithmetic mean and standard deviation.
The geometric mean of the lognormal distribution is . Because the log of a lognormal variable is symmetric and quantiles are preserved under monotonic transformations, the geometric mean of a lognormal distribution is equal to its median.^{[6]}
The geometric mean (m_{g}) can alternatively be derived from the arithmetic mean (m_{a}) in a lognormal distribution by:
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. The correction term can accordingly be 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 geometric standard deviation is equal to .^{[citation needed]}
If X is a lognormally distributed variable, its expected value (E – the arithmetic mean), variance (Var), and standard deviation (s.d.) are
Equivalently, parameters μ and σ can be obtained if the expected value and variance are known; it is simpler if σ is computed first:
For any real or complex number s, the s^{th} moment of lognormal X is given by^{[1]}
A lognormal distribution is not uniquely determined by its moments E[X^{k}] for k ≥ 1, 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 lognormal distribution.^{[citation needed]}
The mode is the point of global maximum of the probability density function. In particular, it solves the equation (ln ƒ)′ = 0:
The median is such a point where F_{X} = 1/2:
The coefficient of variation is the ratio s.d. over m (on the natural scale) and is equal to:
The partial expectation of a random variable X with respect to a threshold k is defined as where is the probability density function of X. Alternatively, and using the definition of conditional expectation, it can be written as g(k)=E[X  X > k]*P(X > k). For a lognormal random variable the partial expectation is given by:
The derivation of the formula is provided in the discussion of this Wikipedia 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.
A set of data that arises from the lognormal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).^{[7]}
The harmonic (H), geometric (G) and arithmetic (A) means of this distribution are related;^{[8]} such relation is given by
Lognormal distributions are infinitely divisible.^{[1]}
The lognormal distribution is important in the description of natural phenomena. The reason is that for many natural processes of growth, growth rate is independent of size. This is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for firms (companies). It can be shown that a growth process following Gibrat's law will result in entity sizes with a lognormal distribution.^{[9]} Examples include:
For determining the maximum likelihood estimators of the lognormal distribution parameters μ and σ, we can use the same procedure as for the normal distribution. To avoid repetition, we observe that
where by ƒ_{L} we denote the probability density function of the lognormal distribution and by ƒ_{N} that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the loglikelihood function thus:
Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, ℓ_{L} and ℓ_{N}, reach their maximum with the same μ and σ. Hence, using the formulas for the normal distribution maximum likelihood parameter estimators and the equality above, we deduce that for the lognormal distribution it holds that
If is a multivariate normal distribution then has a multivariate lognormal distribution^{[22]} with mean
Given a random variate Z drawn from the normal distribution with 0 mean and 1 standard deviation, then the variate
has a lognormal distribution with parameters and .
In the case that all have the same variance parameter , these formulas simplify to
A substitute for the lognormal whose integral can be expressed in terms of more elementary functions^{[24]} can be obtained based on the logistic distribution to get an approximation for the CDF
This is a loglogistic distribution.
Wikimedia Commons has media related to Lognormal distribution. 
