|Probability density function
|Cumulative distribution function
|parameters:||σ2 > 0 — squared scale (real),
μ ∈ R — location
|support:||x ∈ (0, +∞)|
|mgf:||(defined only on the negative half-axis, see text)|
|cf:||representation is asymptotically divergent but sufficient for numerical purposes|
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 a, b ≠ 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.
The probability density function of a log-normal distribution is:
Equivalently, parameters μ and σ can be obtained if the values of mean and variance are known:
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 FX = ½:
If X is distributed log-normally 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 sth moment of log-normal X is given by
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.
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(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
and σ2 < 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 log-normally distributed then Xm is also log-normally 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/(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].
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
This formula has applications in insurance and economics, for example it can be used to derive the Black–Scholes formula.
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:
Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, ℓL ℓL and ℓN, 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
Given a random variate N drawn from the normal distribution with 0 mean and 1 standard deviation, then the variate
has a Log-normal distribution with parameters μ and σ.