Log-normal distribution

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Log-normal
Probability density function
Image:Lognormal distribution PDF.png
μ=0
Cumulative distribution function
Image:Lognormal distribution CDF.png
μ=0
Parameters \sigma \ge 0
-\infty \le \mu \le \infty
Support x \in [0; +\infty)\!
Probability density function (pdf) \frac{1}{x\sigma\sqrt{2\pi}}\exp\left(-\frac{\left[\ln(x)-\mu\right]^2}{2\sigma^2}\right)
Cumulative distribution function (cdf) \frac{1}{2}+\frac{1}{2} \mathrm{erf}\left[\frac{\ln(x)-\mu}{\sigma\sqrt{2}}\right]
Mean e^{\mu+\sigma^2/2}
Median eμ
Mode e^{\mu-\sigma^2}
Variance (e^{\sigma^2}\!\!-1) e^{2\mu+\sigma^2}
Skewness (e^{\sigma^2}\!\!+2)\sqrt{e^{\sigma^2}\!\!-1}
Excess kurtosis \frac{e^{6\sigma^2}-4e^{3\sigma^2}+6e^{\sigma^2}-3}{e^{4\mu+2\sigma^2}(e^{\sigma^2}-1)^4}
Entropy \frac{1}{2}+\frac{1}{2}\ln(2\pi\sigma^2) + \mu
Moment-generating function (mgf) (see text for raw moments)
Characteristic function

In probability and statistics, the log-normal distribution is the single-tailed probability distribution of any random variable whose logarithm is normally distributed. If Y is a random variable with a normal distribution, then X = exp(Y) has a log-normal distribution; likewise, if X is log-normally distributed, then log(X) is normally distributed.

Log-normal is also written log normal or lognormal.

A variable might be modeled as log-normal if it can be thought of as the multiplicative product of many small independent factors. For example the long-term return rate on a stock investment can be considered to be the product of the daily return rates.

Contents

Characterization

Probability density function

The log-normal distribution has the probability density function

f(x;\mu,\sigma) = \frac{e^{-(\ln x - \mu)^2/(2\sigma^2)}}{x \sigma \sqrt{2 \pi}}

for x > 0, where μ and σ are the mean and standard deviation of the variable's logarithm (by definition, the variable's logarithm is normally distributed).

Cumulative distribution function

\frac{1}{2}+\frac{1}{2} \mathrm{erf}\left[\frac{\ln(x)-\mu}{\sigma\sqrt{2}}\right]

Moments

All moments exist and are given by:

\mu_k=e^{k\mu+k^2\sigma^2/2}.

Moment generating function

The moment-generating function does not exist for the log-normal distribution.

Properties

Mean and standard deviation

The expected value is

\mathrm{E}(X) = e^{\mu + \sigma^2/2}

and the variance is

\mathrm{Var}(X) = (e^{\sigma^2} - 1) e^{2\mu + \sigma^2}.\,

Equivalent relationships may be written to obtain μ and σ given the expected value and standard deviation:

\mu = \ln(\mathrm{E}(X))-\frac{1}{2}\ln\left(1+\frac{\mathrm{Var}(X)}{(\mathrm{E}(X))^2}\right),
\sigma^2 = \ln\left(1+\frac{\mathrm{Var}(X)}{(\mathrm{E}(X))^2}\right).

Geometric mean and geometric standard deviation

The geometric mean of the log-normal distribution is exp(μ), and the geometric standard deviation is equal to exp(σ).

If a sample of data is determined to come from a log-normally 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 μ − 3σ \mu_\mathrm{geo} / \sigma_\mathrm{geo}^3
2σ lower bound μ − 2σ \mu_\mathrm{geo} / \sigma_\mathrm{geo}^2
1σ lower bound μ − σ μgeo / σgeo
1σ upper bound μ + σ μgeoσgeo
2σ upper bound μ + 2σ \mu_\mathrm{geo} \sigma_\mathrm{geo}^2
3σ upper bound μ + 3σ \mu_\mathrm{geo} \sigma_\mathrm{geo}^3

Where geometric mean μgeo = exp(μ) and geometric standard deviation σgeo = exp(σ)

Moments

The first few raw moments are:

\mu_1=e^{\mu+\sigma^2/2}
\mu_2=e^{2\mu+4\sigma^2/2}
\mu_3=e^{3\mu+9\sigma^2/2}
\mu_4=e^{4\mu+16\sigma^2/2}

Partial expectation

The partial expectation of a random variable X with respect to a threshold k is defined as

g(k)=\int_k^\infty x f(x)\, dx

where ƒ(x) is the density. For a lognormal density it can be shown that

g(k)=\exp(\mu+\sigma^2/2)\Phi\left(\frac{-\ln(k)+\mu+\sigma^2}{\sigma}\right)

where \scriptstyle\Phi 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).

Maximum likelihood estimation of parameters

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

f_L (x;\mu, \sigma) = \frac 1 x \, f_N (\ln x; \mu, \sigma)

where by f_L (\cdot) we denote the density probability function of the log-normal distribution and by f_N (\cdot)—that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:

\begin{matrix} \ell_L (\mu,\sigma | x_1, x_2, \dots, x_n) & = & - \sum _k \ln x_k + \ell_N (\mu, \sigma | \ln x_1, \ln x_2, \dots, \ln x_n) = \\ \\ \ & = & \operatorname {constant} + \ell_N (\mu, \sigma | \ln x_1, \ln x_2, \dots, \ln x_n). \end{matrix}

Since the first term is constant with regards to μ and σ, both logarithmic likelihood functions, \ell_L and \ell_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 log-normal distribution it holds that

\widehat \mu = \frac {\sum_k \ln x_k} n, \ \widehat \sigma^2 = \frac {\sum_k {\left( \ln x_k - \widehat \mu \right)^2}} n.

Related distributions

  • If X˜N(μ,σ2) is a normal distribution then \exp(X) \sim \operatorname{Log-N}(\mu, \sigma^2).
  • If X_m \sim \operatorname {Log-N} (\mu, \sigma_m^2), \ m = \overline {1 ... n} are independent log-normally distributed variables with the same μ parameter and possibly varying σ, and Y = \prod_{m=1}^n X_m, then Y is a log-normally distributed variable as well: Y \sim \operatorname {Log-N} \left( n\mu, \sum _{m=1}^n \sigma_m^2 \right).
  • Let X_m \sim \operatorname {Log-N} (\mu_m,\sigma_m^2), \ m={1,...,n} \ be independent log-normally distributed variables with

possibly varying σ and μ parameters, and Y=\sum_{m=1}^n X_m. The distribution of Y has no closed-form expression, but can be reasonably approximated by another log-normal distribution Z. A commonly used approximation (due to Fenton and Wilkinson) is obtained by matching the mean and variance:

\sigma^2_Z = \log\left[ \frac{\sum e^{2\mu_m+\sigma_m^2}(e^{\sigma_m^2}-1)}{(\sum e^{\mu_m+\sigma_m^2/2})^2}+1\right]
\mu_Z = \log\left( \sum e^{\mu_m+\sigma_m^2/2} \right)- \frac{\sigma^2_Z}{2}.

In the case that all Xm have the same variance parameter σm = σ, these formulas simplify to

\sigma^2_Z = \log\left[ (e^{\sigma^2}-1)\frac{\sum e^{2\mu_m}}{(\sum e^{\mu_m})^2}+1\right]
\mu_Z = \log\left( \sum e^{\mu_m} \right) + \frac{\sigma^2}{2} - \frac{\sigma^2_Z}{2}.
  • A substitute for the log-normal 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
F(x;\mu,\sigma) = \left[\left(\frac{e^\mu}{x}\right)^{\pi/(\sigma \sqrt{3})} +1\right]^{-1}.

Further reading

References

See also

v • d • e
Probability distributions
de:Logarithmische Normalverteilung

gl:Distribución lognormal it:Variabile casuale logonormalesu:Sebaran Log-normal


Acknowledgement and Attribution Regarding Sources of Content

Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .

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