Probability density function 

Cumulative distribution function 

notation:  

parameters:  σ^{2} > 0 — squared scale (real), μ ∈ R — location 
support:  x ∈ (0, +∞) 
pdf:  
cdf:  
mean:  
median:  
mode:  
variance:  
skewness:  
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 probability distribution of a random variable whose logarithm is normally distributed. If Y is a random variable with a normal distribution, then X = exp(Y) has a lognormal distribution; likewise, if Y is lognormally distributed, then log(Y) is normally distributed. (The base of the logarithmic function does not matter: if log_{a}(Y) is normally distributed, then so is log_{b}(Y), for any two positive numbers a, b ≠ 1.)
Lognormal is also written log normal or lognormal. It is occasionally referred to as the Galton distribution or Galton's distribution.
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. For example, in finance, a longterm discount factor can be derived from the product of shortterm discount factors. In wireless communication, the attenuation caused by shadowing or slow fading from random objects is often assumed to be lognormally distributed. See logdistance path loss model.
Contents 
The probability density function of a lognormal distribution is:
where μ and σ are the mean and standard deviation of the variable’s natural logarithm (by definition, the variable’s logarithm is normally distributed).
where erfc is the complementary error function, and Φ is the standard normal cdf.
If X is a lognormally distributed variable, its expected value (mean), variance, and standard deviation are
Equivalently, parameters μ and σ can be obtained if the values of mean and variance are known:
The geometric mean of the lognormal distribution is e^{μ}, and the geometric standard deviation is equal to e^{σ}.
The mode is the point of global maximum of the pdf function. In particular, it solves the equation (ln ƒ)′ = 0:
The median is such a point where F_{X} = ½:
If X is distributed lognormally with parameters μ and σ, then the (1 − α)confidence interval for X will be
where q* is the (1 − α/2)quantile of the standard normal distribution: q* = Φ^{−1}(1 − α/2).
For any real or complex number s, the s^{th} moment of lognormal X is given by
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.
The characteristic function E[e^{ itX}] has a number of representations. The integral itself converges for Im(t) ≤ 0. The simplest representation is obtained by Taylor expanding e^{ itX} and using formula for moments above.
This series representation is divergent for Re(σ^{2}) > 0, however it is sufficient for numerically evaluating the characteristic function at positive σ as long as the upper limit in sum above is kept bounded, n ≤ N, where
and σ^{2} < 0.1. 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 was derived by Roy Lepnik (see references by this author and by Daniel Dufresne below) by means of double Taylor expansion of e^{(ln x − μ)}2/(2σ^{2}).
The momentgenerating function for the lognormal distribution does not exist on the domain R, but only exists on the halfinterval (−∞, 0].
The partial expectation of a random variable X with respect to a threshold k is defined as g(k) = E[X  X > k]P[X > k]. For a lognormal random variable the partial expectation is given by
This formula has applications in insurance and economics, for example it can be used to derive the Black–Scholes formula.
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} 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
Given a random variate N drawn from the normal distribution with 0 mean and 1 standard deviation, then the variate
has a Lognormal distribution with parameters μ and σ.

