Probability Density Function, A variable X is lognormally distributed if is The general formula for the probability density function of the lognormal distribution is. A random variable X is said to have the lognormal distribution with The lognormal distribution is used to model continuous random quantities when the. Arandom variable X is lognormally distributed if the natural logarithm of X is normally distributed. A lognormal distribution may be specified with.

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In probability theorya log-normal or lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed.

A random variable which is log-normally distributed takes only positive real values. The distribution is occasionally referred to as the Galton distribution or Galton’s distributionafter Francis Galton. A log-normal process is the statistical realization of the multiplicative product of many independent random variableseach 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 distribusii for a random variate X for which the mean and variance of ln X are specified. This relationship is true regardless of the base of the logarithmic or exponential function.

The two sets of parameters can be related as see also Arithmetic moments below [3]. Distrjbusi positive random variable X is log-normally distributed if the logarithm of X is normally distributed. Then we have [1]. The cumulative distribution function is. In consequence distribsi moment generating function is not defined. In consequence, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series. However, a number of alternative divergent series representations have been obtained [5] [6] [7] [8].

A relatively simple approximating formula is available in closed form and given by [9].

Note that the geometric mean is less than the arithmetic mean. This is due to the AM—GM inequalityand corresponds to the logarithm being convex down. For any real or complex number nthe n -th moment of a log-normally distributed variable X is given by [1]. Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are given by.

That is, there exist other distributions with the same set of moments. The mode is the point of global maximum of the probability density function. For a log-normal distribution it is equal to. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.

For a log-normal random variable the partial expectation is given by:. 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.

A set of data that arises from the log-normal distribution has a symmetric Lorenz curve see also Lorenz asymmetry coefficient. Log-normal distributions are infinitely divisible[15] but they are not stable distributionswhich can be easily drawn from.

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 dustribusi the central limit theorem is also known as Gibrat’s lawafter Robert Gibrat — who formulated it for companies.

Even disstribusi that’s not true, the size distributions at any age of things that grow over time tends to be log-normal. Consequently, reference ranges for distribuai in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.

To avoid repetition, we observe that.

### Log Normal Distribution — from Wolfram MathWorld

Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:. 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.

For a more accurate approximation one can use the Monte Carlo method to estimate the cumulative distribution function, the pdf and right tail. From Wikipedia, the free encyclopedia.

Log-normal Probability density function. Lognormal Distributions”, Continuous univariate distributions. Applied Probability and Statistics 2nd ed.

Retrieved 14 April Communications in Statistical — Theory and Methods. Journal of the Optical Society of America. Studies in Applied Mathematics. I — The Characteristic Function”. Thiele report 6 Interpretation and uses of medical statistics 5th ed.

## Log-logistic distribution

International Journal of Mathematics and Mathematical Sciences. Journal of Economic Literature. Silence is also evidence: Proposed Geometric Measures of Accuracy and Precision”. Problems of relative growth. Journal of Chronic Diseases. Retrieved 27 February Diversity and stability in neuronal output rates. Journal of Political Economy. The mis- Behaviour of Markets. An evaluation of different methods”. The European Physical Journal B. Science China Physics, Mechanics and Astronomy. Statistics and Probability Letters.

Bell System Technical Journal. Journal of Hydrologic Engineering. Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf—Mandelbrot. Cauchy lognoemal power Fisher’s z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson’s S U Landau Laplace asymmetric Laplace logistic noncentral t normal Gaussian normal-inverse Gaussian skew normal slash stable Student’s t type-1 Gumbel Tracy—Widom variance-gamma Voigt.

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma logmormal Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart. Degenerate Dirac delta function Singular Cantor. Circular compound Poisson elliptical exponential natural exponential location—scale maximum entropy mixture Pearson Tweedie wrapped.

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### Log-logistic distribution – Wikipedia

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