In probability theory, the Landau distribution is a probability distribution named after Lev Landau.
Because of the distribution's "fat" tail, the moments of the distribution,
like mean or variance, are undefined. The distribution is a particular case of stable distribution.
The stochastic variable is traditionally λ, meaning wavelength.
The original Landau distribution defined by Landau can be evaluated on real numbers as follows:
The Landau distribution, generalized to a stable distribution by introducing position and scale parameters, is as follows:
The relevance of the original definition is as follows:
| stat | λ | note |
|---|---|---|
| mean | N/A | undefined |
| mode | -0.2227829812564085040618242831248... | p(λ)=0.1806556338205509427830338852686... |
| variance | N/A | undefined |
| median | 1.3557804209908013250320928093907... | |
| 0.01-quantile | -2.1048979093493976933783499309591... | |
| 0.05-quantile | -1.4982541517778027339600345356285... | |
| 0.1-quantile | -1.0922545280548463542264694944364... | |
| 0.25-quantile | -0.20464065154575316904929481233852... | |
| 0.75-quantile | 4.45839461019464834851167812598963... | |
| 0.9-quantile | 11.6492846844744055699958678468515... | |
| 0.95-quantile | 22.4502780788727817828880362014437... | |
| 0.99-quantile | 104.156361812207433543595837172678... | |
| entropy | 2.82421914529393668921060013095374... |
The plus λ side is a fat-tail.
The minus λ side decays rapidly.
PDF Precision 64
CDF Precision 64
Quantile Precision 64
Numeric Integration
Asymptotic Expansion
Random Generation
Wolfram Alpha Reference Values
Digits150 source
Digits150 dll
L.Landau, "On the energy loss of fast particles by ionization" (1944)
W.Börsch-Supan, "On the Evaluation of the Function Φ(λ) for Real Values of λ" (1961)
K.S.Kölbig and B.Schorr, "Asymptotic expansions for the Landau density and distribution functions" (1983)
K.S.Kölbig, "On the integral from 0 to infinity of exp(-mu t) t^(nu-1) log(t)^m dt" (1982)