Lognormal distribution
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Probability density function μ=0 

Cumulative distribution function μ=0 

Parameters  σ > 0 

Support  
Probability density function (pdf)  
Cumulative distribution function (cdf)  
Mean  
Median  
Mode  
Variance  
Skewness  
Excess kurtosis  
Entropy  
Momentgenerating function (mgf)  (see text for raw moments) 
Characteristic function  needs your contribution 
In probability and statistics, the lognormal distribution is the singletailed probability distribution of any random variable whose logarithm is normally distributed. If X is a random variable with a normal distribution, then Y = exp(X) 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.
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 
[edit] Characterization
[edit] Probability density function
The lognormal distribution has the probability density function
for x > 0, 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).
[edit] Cumulative distribution function
[edit] Properties
[edit] Mean and standard deviation
If X is a lognormally distributed variable, its expected value (mean) is
and its variance is
hence the standard deviation is
Equivalent relationships may be written to obtain and given the expected value and variance:
[edit] Mode and median
The mode is
The median is
[edit] Geometric mean and geometric standard deviation
The geometric mean of the lognormal distribution is , and the geometric standard deviation is equal to .
If a sample of data is determined to come from a lognormally distributed population, the geometric mean and the geometric standard deviation may be used to estimate confidence intervals akin to the way the arithmetic mean and standard deviation are used to estimate confidence intervals for a normally distributed sample of data.
Confidence interval bounds  log space  geometric 

3σ lower bound  
2σ lower bound  
1σ lower bound  
1σ upper bound  
2σ upper bound  
3σ upper bound 
Where geometric mean and geometric standard deviation
[edit] Moments
For any real number s, the s^{th} moment 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.
[edit] Moment generating function
The momentgenerating function for the lognormal distribution does not exist on the domain R, but the moment generating function does exist on (−∞, 0]. The set {t : g(t) < ∞}, where g is the momentgenerating function, contains (−∞, 0].
[edit] Partial expectation
The partial expectation of a random variable X with respect to a threshold k is defined as
where is the density. For a lognormal density it can be shown that
where is the cumulative distribution function of the standard normal. The partial expectation of a lognormal has applications in insurance and in economics (for example it can be used to derive the Black–Scholes formula).
[edit] Maximum likelihood estimation of parameters
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, and , 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
[edit] Generating Lognormallydistributed random variates
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.
[edit] Related distributions
 If is a normal distribution then .
 If is a lognormally distributed random variable then ln(Y)˜N(μ,σ^{2}) is a normally distributed random variable.
 If are independent lognormally distributed variables with the same μ parameter and possibly varying σ, and , then Y is a lognormally distributed variable as well:
 Let be independent lognormally distributed variables with
possibly varying σ and μ parameters, and . The distribution of Y has no closedform expression, but can be reasonably approximated by another lognormal distribution Z at the right tail. Its probability density function at the neighborhood of 0 is characterized in (Gao et al., 2009) and it does not resemble any lognormal distribution. A commonly used approximation (due to Fenton and Wilkinson) is obtained by matching the mean and variance:
In the case that all X_{m} have the same variance parameter σ_{m} = σ, these formulas simplify to
 A substitute for the lognormal whose integral can be expressed in terms of more elementary functions (Swamee, 2002) can be obtained based on the logistic distribution to get the CDF
This is a loglogistic distribution.
 If then X + c is called shifted lognormal.
[edit] Further reading
 Robert Brooks, Jon Corson, and J. Donal Wales. "The Pricing of Index Options When the Underlying Assets All Follow a Lognormal Diffusion", in Advances in Futures and Options Research, volume 7, 1994.
[edit] References
 The Lognormal Distribution, Aitchison, J. and Brown, J.A.C. (1957)
 Lognormal Distributions across the Sciences: Keys and Clues, E. Limpert, W. Stahel and M. Abbt,. BioScience, 51 (5), p. 341–352 (2001).
 Eric W. Weisstein et al. Log Normal Distribution at MathWorld. Electronic document, retrieved October 26, 2006.
 Swamee, P.K. (2002). Near Lognormal Distribution, Journal of Hydrologic Engineering. 7(6): 441444
 Roy B. Leipnik (1991), On Lognormal Random Variables: I  The Characteristic Function, Journal of the Australian Mathematical Society Series B, vol. 32, pp 327–347.
 Gao et al. (2009) [1], Asymptotic Behaviors of Tail Density for Sum of Correlated Lognormal Variables". International Journal of Mathematics and Mathematical Sciences.
[edit] See also
 Normal distribution
 Geometric mean
 Geometric standard deviation
 Error function
 Logdistance path loss model
 Slow fading