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Notation ln N ( μ , σ 2 ) {displaystyle ln {mathcal {N}}(mu ,,sigma ^{2})} Parameters μ ∈ R {displaystyle mu in mathbb {R} } — location, σ > 0 {displaystyle sigma >0} — scaleof associated normal Support x ∈ ( 0 , + ∞ ) {displaystyle xin (0,+infty )} PDF 1 x σ 2 π e − ( ln x − μ ) 2 2 σ 2 {displaystyle {rac {1}{xsigma {sqrt {2pi }}}} e^{-{rac {left(ln x-mu ight)^{2}}{2sigma ^{2}}}}} CDF 1 2 + 1 2 e r f [ ln x − μ 2 σ ] {displaystyle {rac {1}{2}}+{rac {1}{2}},mathrm {erf} {Big [}{rac {ln x-mu }{{sqrt {2}}sigma }}{Big ]}} Mean e μ + σ 2 / 2 {displaystyle e^{mu +sigma ^{2}/2}} |
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable
Contents
- Notation
- Probability density function
- Cumulative distribution function
- Characteristic function and moment generating function
- Location and scale
- Geometric moments
- Arithmetic moments
- Mode and median
- Arithmetic coefficient of variation
- Partial expectation
- Conditional expectation
- Other
- Occurrence and applications
- Extremal principle of entropy to fix the free parameter
- Maximum likelihood estimation of parameters
- Multivariate log normal
- Related distributions
- References
A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain. The log-normal distribution is the maximum entropy probability distribution for a random variate
Notation
Given a log-normally distributed random variable
with
This relationship is true regardless of the base of the logarithmic or exponential function. If
On a logarithmic scale,
In contrast, the mean, standard deviation, and variance of the non-logarithmized sample values are respectively denoted
Probability density function
A random positive variable
A change of variables must conserve differential probability. In particular,
where
is the log-normal probability density function.
Cumulative distribution function
The cumulative distribution function is
where erfc is the complementary error function, and Φ is the cumulative distribution function of the standard normal distribution.
Characteristic function and moment generating function
All moments of the log-normal distribution exist and
This can be derived by letting
The characteristic function
However, a number of alternative divergent series representations have been obtained
A closed-form formula for the characteristic function
where
Location and scale
The location and scale parameters of a log-normal distribution, i.e.
Geometric moments
The geometric mean of the log-normal distribution is
Because the log-transformed variable
Note that the geometric mean is less than the arithmetic mean. This is due to the AM–GM inequality, and corresponds to the logarithm being convex down. In fact,
In finance the term
Arithmetic moments
For any real or complex number s, the s-th moment of a log-normally distributed variable X is given by
Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are given by
respectively.
The location (μ) and scale (σ) parameters can be obtained if the arithmetic mean and the arithmetic variance are known:
A probability distribution is not uniquely determined by the moments E[Xk] = ekμ + 1/2k2σ2 for k ≥ 1. That is, there exist other distributions with the same set of moments. In fact, there is a whole family of distributions with the same moments as the log-normal distribution.
Mode and median
The mode is the point of global maximum of the probability density function. In particular, it solves the equation
The median is such a point where
Arithmetic coefficient of variation
The arithmetic coefficient of variation
Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.
Partial expectation
The partial expectation of a random variable
where
where Φ is the normal cumulative distribution function. The derivation of the formula is provided in the discussion of this Wikipedia entry. The partial expectation formula has applications in insurance and economics, it is used in solving the partial differential equation leading to the Black–Scholes formula.
Conditional expectation
The conditional expectation of a lognormal random variable X with respect to a threshold k is its partial expectation divided by the cumulative probability of being in that range:
Other
A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).
The harmonic
Log-normal distributions are infinitely divisible, but they are not stable distributions, which can be easily drawn from.
Occurrence and applications
The log-normal distribution is important in the description of natural phenomena. This follows, because many natural growth processes are driven by the accumulation of many small percentage changes. These become additive on a log scale. If the effect of any one change is negligible, the central limit theorem says that the distribution of their sum is more nearly normal than that of the summands. When back-transformed onto the original scale, it makes the distribution of sizes approximately log-normal (though if the standard deviation is sufficiently small, the normal distribution can be an adequate approximation).
This multiplicative version of the central limit theorem is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies. If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if that's not true, the size distributions at any age of things that grow over time tends to be log-normal.
Examples include the following:
Consequently, reference ranges for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.
Extremal principle of entropy to fix the free parameter
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
where by
Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions,
Multivariate log-normal
If