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Log-normal distribution

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 X is log-normally distributed, then Y = ln X has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y, X =…

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CDF
1 2 [ 1 + erf ⁡ ( ln ⁡ x − μ σ 2 ) ] = Φ ( ln ⁡ x − μ σ ) {\displaystyle {\begin{aligned}&{\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {\ln x-\mu }{\sigma {\sqrt {2}}}…
CF
representation ∑ n = 0 ∞ ( i t ) n n ! e n μ + n 2 σ 2 / 2 {\displaystyle \sum _{n=0}^{\infty }{\frac {{\left(it\right)}^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}} is asymptotically…
Entropy
log 2 ⁡ ( 2 π e σ e μ ) {\displaystyle \log _{2}\left({\sqrt {2\pi e}}\,\sigma e^{\mu }\right)}
Excess kurtosis
exp ⁡ ( 4 σ 2 ) + 2 exp ⁡ ( 3 σ 2 ) + 3 exp ⁡ ( 2 σ 2 ) − 6 {\displaystyle \exp \left(4\sigma ^{2}\right)+2\exp \left(3\sigma ^{2}\right)+3\exp \left(2\sigma ^{2}\right)-6}
Fisher information
1 σ 2 ( 1 0 0 2 ) {\displaystyle {\frac {1}{\sigma ^{2}}}{\begin{pmatrix}1&0\\0&2\end{pmatrix}}}
Mean
exp ⁡ ( μ + σ 2 2 ) {\displaystyle \exp \left(\mu +{\frac {\sigma ^{2}}{2}}\right)}

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Log-normal distribution

Nodes126
Edges125
Triples134
Avg. degree1.98
Density0.015873
Components1

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Log-normal distribution

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related to Biology and medicine · 15
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related to Social sciences and demographics · 15
Log-normal distribution → Benoit Mandelbrot, Black, City, From, Gibrat's Law, Gini, However, If, In, Indeed, Lévy, Pareto, Phi, Scholes, The
has application · 12
Log-normal distribution → Assuming, Consequently, Even, Examples, Gibrat's, If, Many, Multiplicative Central Limit Theorem, Robert Gibrat, The, This, Under
related to Related distributions · 11
Log-normal distribution → Fenton, For, If, In, Its, Let, Lognormal, Marlow, Monte Carlo, The, Wilkinson
related to Technology · 11
Log-normal distribution → Also, File, Gaussian, In, Internet, MIME, Particle, Sizes, The, This, Windows OS
related to Geometric or multiplicative moments · 10
Log-normal distribution → By, Coefficient, CV, GCV, GM, GSD, GVar, It, The, This
related to Alternative parameterizations · 9
Log-normal distribution → In, LogNormal1, LogNormal2, LogNormal3, LogNormal4, LogNormal5, LogNormal6, LogNormal7, ProbOnto
related to Extremal principle of entropy to fix the free parameter σ · 5
Log-normal distribution → For, In, Shannon, These, This
related to Human behavior · 4
Log-normal distribution → Internet, Onset, The, Users
related to Physical sciences · 4
Log-normal distribution → In, Polarstern, Southern Atlantic Ocean, The

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Important terminology

displaystyle distribution log-normal sigma mu operatorname ln frac right left mean normal distributions sqrt function random lognormal distributed parameters exp

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Log-normal distributionCDF1 2 [ 1 + erf ⁡ ( ln ⁡ x − μ σ 2 ) ] = Φ ( ln ⁡ x − μ σ ) {\displaystyle {\begin{aligned}&{\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {\ln x-\mu }{\sigma {\sqrt {2}}}…1.00infobox
Log-normal distributionCFrepresentation ∑ n = 0 ∞ ( i t ) n n ! e n μ + n 2 σ 2 / 2 {\displaystyle \sum _{n=0}^{\infty }{\frac {{\left(it\right)}^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}} is asymptotically…1.00infobox
Log-normal distributionEntropylog 2 ⁡ ( 2 π e σ e μ ) {\displaystyle \log _{2}\left({\sqrt {2\pi e}}\,\sigma e^{\mu }\right)}1.00infobox
Log-normal distributionExcess kurtosisexp ⁡ ( 4 σ 2 ) + 2 exp ⁡ ( 3 σ 2 ) + 3 exp ⁡ ( 2 σ 2 ) − 6 {\displaystyle \exp \left(4\sigma ^{2}\right)+2\exp \left(3\sigma ^{2}\right)+3\exp \left(2\sigma ^{2}\right)-6}1.00infobox
Log-normal distributionFisher information1 σ 2 ( 1 0 0 2 ) {\displaystyle {\frac {1}{\sigma ^{2}}}{\begin{pmatrix}1&0\\0&2\end{pmatrix}}}1.00infobox
Log-normal distributionMeanexp ⁡ ( μ + σ 2 2 ) {\displaystyle \exp \left(\mu +{\frac {\sigma ^{2}}{2}}\right)}1.00infobox
Log-normal distributionMedianexp ⁡ ( μ ) {\displaystyle \exp(\mu )}1.00infobox
Log-normal distributionMethod of momentsμ = ln ⁡ E ⁡ [ X ] − 1 2 ln ⁡ ( Var ⁡ [ X ] E ⁡ [ X ] 2 + 1 ) , {\displaystyle \mu =\ln \operatorname {E} [X]-{\frac {1}{2}}\ln \left({\frac {\operatorname {Var} [X]}{\operatorn…1.00infobox
Log-normal distributionMGFdefined only for numbers with a non-positive real part, see text1.00infobox
Log-normal distributionModeexp ⁡ ( μ − σ 2 ) {\displaystyle \exp \left(\mu -\sigma ^{2}\right)}1.00infobox
Log-normal distributionNotationLognormal ⁡ ( μ , σ 2 ) {\displaystyle \operatorname {Lognormal} \left(\mu ,\,\sigma ^{2}\right)}1.00infobox
Log-normal distributionParametersμ ∈ ( − ∞ , + ∞ ) {\displaystyle \mu \in (-\infty ,+\infty )} (logarithm of location),1.00infobox
Log-normal distributionParametersσ > 0 {\displaystyle \sigma >0} (logarithm of scale)1.00infobox
Log-normal distributionPDF1 x σ 2 π exp ⁡ ( − ( ln ⁡ x − μ ) 2 2 σ 2 ) {\displaystyle {\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {\left(\ln x-\mu \right)^{2}}{2\sigma ^{2}}}\right)}1.00infobox
Log-normal distributionQuantileexp ⁡ ( μ + 2 σ 2 erf − 1 ⁡ ( 2 p − 1 ) ) = exp ⁡ ( μ + σ Φ − 1 ( p ) ) {\displaystyle {\begin{aligned}&\exp \left(\mu +{\sqrt {2\sigma ^{2}}}\operatorname {erf} ^{-1}(2p-1)\rig…1.00infobox
Log-normal distributionSkewness[ exp ⁡ ( σ 2 ) + 2 ] exp ⁡ ( σ 2 ) − 1 {\displaystyle \left[\exp \left(\sigma ^{2}\right)+2\right]{\sqrt {\exp(\sigma ^{2})-1}}}1.00infobox
Log-normal distributionSupportx ∈ ( 0 , + ∞ ) {\displaystyle x\in (0,+\infty )}1.00infobox
Log-normal distributionVariance[ exp ⁡ ( σ 2 ) − 1 ] exp ⁡ ( 2 μ + σ 2 ) {\displaystyle \left[\exp(\sigma ^{2})-1\right]\exp \left(2\mu +\sigma ^{2}\right)}1.00infobox
Log-normal distributionis amaximum entropy probability distribution for a random variate X0.90text
Log-normal distributionis aspecial case of the semi-bounded Johnson's SU-distribution.If X0.90text

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