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Encyclopedia > Rayleigh distribution
Rayleigh
Probability density function
Plot of the Rayleigh PDF
Cumulative distribution function
Plot of the Rayleigh CDF
Parameters sigma>0,
Support xin [0;infty)
Probability density function (pdf) frac{x expleft(frac{-x^2}{2sigma^2}right)}{sigma^2}
Cumulative distribution function (cdf) 1-expleft(frac{-x^2}{2sigma^2}right)
Mean sigma sqrt{frac{pi}{2}}
Median sigmasqrt{ln(4)},
Mode sigma,
Variance frac{4 - pi}{2} sigma^2
Skewness frac{2sqrt{pi}(pi - 3)}{(4-pi)^{3/2}}
Excess kurtosis -frac{6pi^2 - 24pi +16}{(4-pi)^2}
Entropy 1+lnleft(frac{1}{sqrt{2}sigma^3}right)+frac{gamma}{2}
Moment-generating function (mgf) 1+sigma t,e^{sigma^2t^2/2}sqrt{frac{pi}{2}} left(textrm{erf}left(frac{sigma t}{sqrt{2}}right)!+!1right)
Characteristic function 1!-!sigma te^{-sigma^2t^2/2}sqrt{frac{pi}{2}}!left(textrm{erfi}!left(frac{sigma t}{sqrt{2}}right)!-!iright)

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. It usually arises when a two-dimensional vector (e.g. wind velocity) has its two orthogonal components normally and independently distributed. The absolute value (e.g. wind speed) will then have a Rayleigh distribution. The distribution may also arise in the case of random complex numbers whose real and imaginary components are normally and independently distributed. The absolute value of these numbers will then be Rayleigh-distributed. Image File history File links Download high resolution version (1300x975, 156 KB) Probability density function for the Rayleigh distribution File links The following pages link to this file: Rayleigh distribution ... Image File history File links Download high resolution version (1300x975, 166 KB) Cumulative density function for the Rayleigh distribution File links The following pages link to this file: Rayleigh distribution ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... It has been suggested that this article or section be merged with Probability axioms. ... A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... This article or section does not adequately cite its references or sources. ... The velocity of an object is its speed in a particular direction. ... The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)) is a probability distribution of great importance in many fields. ... This article or section does not adequately cite its references or sources. ...


The probability density function is

f(x|sigma) = frac{x expleft(frac{-x^2}{2sigma^2}right)}{sigma^2}.

The characteristic function is given by: In probability theory, the characteristic function of any random variable completely defines its probability distribution. ...

varphi(t)=
1!-!sigma te^{-sigma^2t^2/2}sqrt{frac{pi}{2}}!left(textrm{erfi}!left(frac{sigma t}{sqrt{2}}right)!-!iright)

where operatorname{erfi}(z) is the complex error function. The moment generating function is given by Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ... In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith...

M(t)=,
1+sigma t,e^{sigma^2t^2/2}sqrt{frac{pi}{2}} left(textrm{erf}left(frac{sigma t}{sqrt{2}}right)!+!1right),

where erf(z) is the error function. The raw moments are then given by Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ... -1...

mu_k=sigma^k2^{k/2},Gamma(1+k/2),

where Γ(z) is the Gamma function. The moments may be used to calculate: The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ...


Mean: sigma sqrt{frac{pi}{2}}


Variance: frac{4-pi}{2} sigma^2


Skewness: frac{2sqrt{pi}(pi - 3)}{(4-pi)^{3/2}}


Kurtosis: - frac{6pi^2 - 24pi +16}{(4-pi)^2}


Parameter estimation

Given N independent and identically distributed Rayleigh random variables with parameter σ, the maximum likelihood estimate of σ is Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ...

hat{sigma}=sqrt{frac{1}{2N}sum_{i=1}^N x_i^2}.

Related distributions

The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)) is a probability distribution of great importance in many fields. ... In probability theory and statistics, the chi-square distribution (also chi-squared or χ2  distribution) is one of the theoretical probability distributions most widely used in inferential statistics, i. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions that represents the sum of exponentially distributed random variables, each of which has mean . ... In probability theory and statistics, the chi distribution is a continuous probability distribution. ... In probability theory and statistics, the Rice distribution distribution is a continuous probability distribution. ... In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ...

See also

Image:Bvn-small.png Probability distributions [ view • talk • edit ]
Univariate Multivariate
Discrete: Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot Ewens • multinomial • multivariate Polya
Continuous: Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square)• inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda Dirichlet • inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling • singular

  Results from FactBites:
 
RR-4121 : Modelling SAR with a Generalisation of the Rayleigh Distribution (556 words)
some alternative distributions have been suggested such as weibull and log-normal distributions, however, in most of the cases these models are empirical, not derived with the consideration of underlying physical conditions and therefore are case specific.
we also derive novel methods for the estimation of the heavy-tailed rayleigh distribution parameter- s based on negative fractional-order statistics for model fitting.
Nous présentons également de nouvelles mèthodes d'estimation des paramètres d'une distribution de Rayleigh à queue lourde fondées sur des statistiques d'ordre fractionnaire négatif.
Rayleigh distribution - Wikipedia, the free encyclopedia (300 words)
The Chi distribution is a generalization of the Rayleigh distribution.
The Rice distribution is a generalization of the Rayleigh distribution.
The Weibull distribution is a generalization of the Rayleigh distribution.
  More results at FactBites »


 

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