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Log-normal
Probability density function
Plot of the Lognormal PMF
Cumulative distribution function
Plot of the Lognormal CMF
notation: \ln\mathcal{N}(\mu,\,\sigma^2)
parameters: σ2 > 0 — squared scale (real),
μR — location
support: x ∈ (0, +∞)
pdf: \frac{1}{x\sqrt{2\pi\sigma^2}}\, e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}}
cdf: \frac12 + \frac12\,\mathrm{erf}\Big[\frac{\ln x-\mu}{\sqrt{2\sigma^2}}\Big]
mean: e^{\mu+\sigma^2/2}
median: e^{\mu}\,
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}
kurtosis: e^{4\sigma^2}\!\! + 2e^{3\sigma^2}\!\! + 3e^{2\sigma^2}\!\! - 3
entropy: \frac12 + \frac12 \ln(2\pi\sigma^2) + \mu
mgf: (defined only on the negative half-axis, see text)
cf: representation \sum_{n=0}^{\infty}\frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2} is asymptotically divergent but sufficient for numerical purposes
Fisher information: \begin{pmatrix}1/\sigma^2&0\\0&1/(2\sigma^4)\end{pmatrix}

In probability theory, a log-normal 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 log-normal distribution; likewise, if Y is log-normally distributed, then log(Y) is normally distributed. (The base of the logarithmic function does not matter: if loga(Y) is normally distributed, then so is logb(Y), for any two positive numbers ab ≠ 1.)

Log-normal 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 log-normal 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 long-term discount factor can be derived from the product of short-term discount factors. In wireless communication, the attenuation caused by shadowing or slow fading from random objects is often assumed to be log-normally distributed. See log-distance path loss model.

Contents

Characterization

Probability density function

The probability density function of a log-normal distribution is:

f_X(x;\mu,\sigma) = \frac{1}{x \sigma \sqrt{2 \pi}}\, e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}},\ \ 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).

Cumulative distribution function

F_X(x;\mu,\sigma) = \frac12 \operatorname{erfc}\!\left[-\frac{\ln x - \mu}{\sigma\sqrt{2}}\right] = \Phi\bigg(\frac{\ln x - \mu}{\sigma}\bigg),

where erfc is the complementary error function, and Φ is the standard normal cdf.

Mean and standard deviation

If X is a lognormally distributed variable, its expected value (mean), variance, and standard deviation are

\begin{align} & \mathrm{E}[X] = e^{\mu + \tfrac{1}{2}\sigma^2}, \ & \mathrm{Var}[X] = (e^{\sigma^2} - 1) e^{2\mu + \sigma^2} \ & \mathrm{s.d}[X] = \sqrt{\mathrm{Var}[X]} = e^{\mu + \tfrac{1}{2}\sigma^2}\sqrt{e^{\sigma^2} - 1} \end{align}

Equivalently, parameters μ and σ can be obtained if the values of mean and variance are known:

\begin{align} \mu &= \ln(\mathrm{E}[X]) - \frac12 \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). \end{align}

The geometric mean of the log-normal distribution is eμ, and the geometric standard deviation is equal to eσ.

Mode and median

The mode is the point of global maximum of the pdf function. In particular, it solves the equation (ln ƒ)′ = 0:

\mathrm{Mode}[X] = e^{\mu - \sigma^2}.

The median is such a point where FX = ½:

\mathrm{Med}[X] = e^{\mu}\,.

Confidence interval

If X is distributed log-normally with parameters μ and σ, then the (1 − α)-confidence interval for X will be

X \in \Big[\, e^{\mu - \sigma q^*}\!,\ e^{\mu + \sigma q^*} \,\Big],

where q* is the (1 − α/2)-quantile of the standard normal distribution: q* = Φ−1(1 − α/2).

Moments

For any real or complex number s, the sth moment of log-normal X is given by

\operatorname{E}[X^s] = e^{s\mu + \tfrac{1}{2}s^2\sigma^2}.

A log-normal distribution is not uniquely determined by its moments E[Xk] for k ≥ 1, that is, there exists some other distribution with the same moments for all k.

Characteristic function and moment generating function

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.

\varphi(t) = \sum_{n=0}^\infty \frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2}.

This series representation is divergent for Re(σ2) > 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

\max(|t|,|\mu|) \ll N \ll \frac{2}{\sigma^2}\ln\frac{2}{\sigma^2}

and σ2 < 0.1. To bring the numerical values of parameters μσ into the domain where strong inequality holds true one could use the fact that if X is log-normally distributed then Xm is also log-normally distributed with parameters μmσm. Since  \mu\sigma^2 \propto m^3, 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/(2σ2).

The moment-generating function for the log-normal distribution does not exist on the domain R, but only exists on the half-interval (−∞, 0].

Partial expectation

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 log-normal random variable the partial expectation is given by

g(k) = \int_k^\infty \!xf(x)\, dx = e^{\mu+\tfrac{1}{2}\sigma^2}\, \Phi\!\left(\frac{\mu+\sigma^2-\ln k}{\sigma}\right).

This formula has applications in insurance and 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) = \prod_{i=1}^n \left(\frac 1 x_i\right) \, f_N (\ln x; \mu, \sigma)

where by ƒL we denote the probability density function of the log-normal distribution and by ƒN that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:

 \begin{align} \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{align}

Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, L L and 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}.

Generating log-normally-distributed random variates

Given a random variate N drawn from the normal distribution with 0 mean and 1 standard deviation, then the variate

X= e^{\mu + \sigma N}\,

has a Log-normal distribution with parameters μ and σ.

Related distributions

  • If X \sim \mathcal{N}(\mu, \sigma^2) is a normal distribution, then \exp(X) \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2).
  • If X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2) is distributed log-normally, then \ln(X) \sim \mathcal{N}(\mu, \sigma^2) is a normal random variable.
  • If X_j \sim \operatorname{Log-\mathcal{N}}(\mu_j, \sigma_j^2) are n independent log-normally distributed variables, and Y = \textstyle\prod_{j=1}^n X_j, then Y is also distributed log-normally:
    Y \sim \operatorname{Log-\mathcal{N}}\Big(\textstyle \sum_{j=1}^n\mu_j,\ \sum_{j=1}^n \sigma_j^2 \Big).
  • Let X_j \sim \operatorname{Log-\mathcal{N}}(\mu_j,\sigma_j^2)\ be independent log-normally distributed variables with possibly varying σ and μ parameters, and Y=\textstyle\sum_{j=1}^n X_j. The distribution of Y has no closed-form expression, but can be reasonably approximated by another log-normal 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 log-normal distribution. A commonly used approximation (due to Fenton and Wilkinson) is obtained by matching the mean and variance:
    \begin{align} \sigma^2_Z &= \log\!\left[ \frac{\sum e^{2\mu_j+\sigma_j^2}(e^{\sigma_j^2}-1)}{(\sum e^{\mu_j+\sigma_j^2/2})^2} + 1\right], \ \mu_Z &= \log\!\left[ \sum e^{\mu_j+\sigma_j^2/2} \right] - \frac{\sigma^2_Z}{2}. \end{align}
    In the case that all Xj have the same variance parameter σj = σ, these formulas simplify to
    \begin{align} \sigma^2_Z &= \log\!\left[ (e^{\sigma^2}-1)\frac{\sum e^{2\mu_j}}{(\sum e^{\mu_j})^2} + 1\right], \ \mu_Z &= \log\!\left[ \sum e^{\mu_j} \right] + \frac{\sigma^2}{2} - \frac{\sigma^2_Z}{2}. \end{align}
  • If X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2), then X + c is said to have a shifted log-normal distribution. E[X + c] = E[X] + c, Var[X + c] = Var[X].
  • If X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2), then Y = aX is also log-normal, Y \sim \operatorname{Log-\mathcal{N}}( \ln a+\mu,\ \sigma^2).
  • If X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2), then Y = 1X is also log-normal, Y \sim \operatorname{Log-\mathcal{N}}(-\mu,\ \sigma^2).

Similar distributions

  • 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}.
This is a log-logistic distribution.
  • An exGaussian distribution is the distribution of the sum of a normally distributed random variable and an exponentially distributed random variable. This has a similar long tail, and has been used as a model for reaction times.

Further reading

References

See also








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