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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the Rayleigh distribution is a
continuous probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
for nonnegative-valued
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s. Up to rescaling, it coincides with the
chi distribution In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard norm ...
with two degrees of freedom. The distribution is named after
Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Amo ...
(). A Rayleigh distribution is often observed when the overall magnitude of a vector is related to its directional
components Circuit Component may refer to: •Are devices that perform functions when they are connected in a circuit.   In engineering, science, and technology Generic systems * System components, an entity with discrete structure, such as an assem ...
. One example where the Rayleigh distribution naturally arises is when
wind Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few hou ...
velocity is analyzed in
two dimensions In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
. Assuming that each component is
uncorrelated In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, there ...
, normally distributed with equal
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
, and zero
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithme ...
, then the overall wind speed (
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
magnitude) will be characterized by a Rayleigh distribution. A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are independently and identically distributed
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.


Definition

The
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
of the Rayleigh distribution isPapoulis, Athanasios; Pillai, S. (2001) ''Probability, Random Variables and Stochastic Processes''. , :f(x;\sigma) = \frac e^, \quad x \geq 0, where \sigma is the
scale parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family o ...
of the distribution. The
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
is :F(x;\sigma) = 1 - e^ for x \in bivariate_normally_distributed,_centered_at_zero,_and_independent.__Then_U_and_V_have_density_functions :f_U(x;_\sigma)_=_f_V(x;\sigma)_=_\frac. Let_X_be_the_length_of_Y._That_is,_X_=_\sqrt.__Then_X_has_cumulative_distribution_function :F_X(x;_\sigma)_=_\iint__f_U(u;\sigma)_f_V(v;\sigma)_\,dA, where_D_x_is_the_disk :D_x_=_\left\. Writing_the_
bivariate_normally_distributed,_centered_at_zero,_and_independent.__Then_U_and_V_have_density_functions :f_U(x;_\sigma)_=_f_V(x;\sigma)_=_\frac. Let_X_be_the_length_of_Y._That_is,_X_=_\sqrt.__Then_X_has_cumulative_distribution_function :F_X(x;_\sigma)_=_\iint__f_U(u;\sigma)_f_V(v;\sigma)_\,dA, where_D_x_is_the_disk :D_x_=_\left\. Writing_the_multiple_integral">double_integral_ In_mathematics_(specifically_multivariable_calculus),_a_multiple_integral_is_a_definite_integral_of_a_function_of_several_real_variables,_for_instance,__or_._Integrals_of_a_function_of_two_variables_over_a_region_in_\mathbb^2_(the__real-number__...
_in_polar_coordinate_system.html" "title="multiple_integral.html" "title="bivariate normal distribution">bivariate normally distributed, centered at zero, and independent. Then U and V have density functions :f_U(x; \sigma) = f_V(x;\sigma) = \frac. Let X be the length of Y. That is, X = \sqrt. Then X has cumulative distribution function :F_X(x; \sigma) = \iint_ f_U(u;\sigma) f_V(v;\sigma) \,dA, where D_x is the disk :D_x = \left\. Writing the multiple integral">double integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
in polar coordinate system">polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
, it becomes :F_X(x; \sigma) = \frac \int_0^ \int_0^x r e^ \,dr\,d\theta = \frac 1 \int_0^x r e^ \,dr. Finally, the probability density function for X is the derivative of its cumulative distribution function, which by the fundamental theorem of calculus is :f_X(x;\sigma) = \frac d F_X(x;\sigma) = \frac x e^, which is the Rayleigh distribution. It is straightforward to generalize to vectors of dimension other than 2. There are also generalizations when the components have
unequal variance In statistics, a sequence (or a vector) of random variables is homoscedastic () if all its random variables have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The s ...
or correlations (
Hoyt distribution The Nakagami distribution or the Nakagami-''m'' distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter m\geq 1/2 and a second parameter controlling ...
), or when the vector ''Y'' follows a bivariate Student ''t''-distribution (see also:
Hotelling's T-squared distribution In statistics, particularly in hypothesis testing, the Hotelling's ''T''-squared distribution (''T''2), proposed by Harold Hotelling, is a multivariate probability distribution that is tightly related to the ''F''-distribution and is most not ...
). Suppose Y is a random vector with components u,v that follows a
multivariate t-distribution In statistics, the multivariate ''t''-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's ''t''-distribution, which is a distribution applicab ...
. If the components both have mean zero, equal variance, and are independent, the bivariate Student's-t distribution takes the form: :f(u,v) = \left( 1 + \right)^ Let R = \sqrt be the magnitude of Y. Then the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
(CDF) of the magnitude is: : F(r) = \iint_ \left( 1 + \right)^du \; dv where D_ is the disk defined by: : D_ = \left\ Converting to
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
leads to the CDF becoming: : \begin F(r) &= \int_^\int_^ \rho\left( 1 + \right)^d\theta \; d\rho \\ &= \int_^\rho\left( 1 + \right)^ d\rho \\ &= 1-\left( 1 + \right)^ \end Finally, the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
(PDF) of the magnitude may be derived: : f(r) = F'(r) = \left( 1 + \right)^ In the limit as \nu \rightarrow \infty , the Rayleigh distribution is recovered because: : \lim_ \left( 1 + \right)^ = e^


Properties

The raw moments are given by: : \mu_j = \sigma^j2^\,\Gamma\left(1 + \frac j 2\right), where \Gamma(z) is the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. The
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithme ...
of a Rayleigh random variable is thus : :\mu(X) = \sigma \sqrt\ \approx 1.253\ \sigma. The
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
of a Rayleigh random variable is: :\operatorname(X) = \sqrt \sigma \approx 0.655\ \sigma The
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
of a Rayleigh random variable is : :\operatorname(X) = \mu_2-\mu_1^2 = \left(2-\frac\right) \sigma^2 \approx 0.429\ \sigma^2 The
mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...
is \sigma, and the maximum pdf is : f_ = f(\sigma;\sigma) = \frac e^ \approx \frac. The
skewness In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal d ...
is given by: :\gamma_1 = \frac \approx 0.631 The excess
kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
is given by: :\gamma_2 = -\frac \approx 0.245 The
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
is given by: :\varphi(t) = 1 - \sigma te^\sqrt \left operatorname\left(\frac\right) - i\right/math> where \operatorname(z) is the imaginary
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non-elementary ...
. The
moment generating function In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
is given by : M(t) = 1 + \sigma t\,e^\sqrt \left operatorname\left(\frac\right) + 1\right/math> where \operatorname(z) is the
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non-elementary ...
.


Differential entropy

The
differential entropy Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Claude Shannon to extend the idea of (Shannon) entropy, a measure of average surprisal of a random variable, to continu ...
is given by :H = 1 + \ln\left(\frac \sigma \right) + \frac \gamma 2 where \gamma is the
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...
.


Parameter estimation

Given a sample of ''N''
independent and identically distributed In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usua ...
Rayleigh random variables x_i with parameter \sigma, : \widehat^2 = \!\,\frac\sum_^N x_i^2 is the
maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...
estimate and also is
unbiased Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
. :\widehat\approx \sqrt is a biased estimator that can be corrected via the formula :\sigma = \widehat \frac = \widehat \frac


Confidence intervals

To find the (1 − ''α'') confidence interval, first find the bounds ,b/math> where: :  P(\chi_^2 \leq a) = \alpha/2, \quad P(\chi_^2 \leq b) = 1 - \alpha/2 then the scale parameter will fall within the bounds :  \frac \leq ^2 \leq \frac


Generating random variates

Given a random variate ''U'' drawn from the uniform distribution in the interval (0, 1), then the variate :X=\sigma\sqrt\, has a Rayleigh distribution with parameter \sigma. This is obtained by applying the
inverse transform sampling Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, or the golden ruleAalto University, N. Hyvönen, Computational methods in inverse probl ...
-method.


Related distributions

* R \sim \mathrm(\sigma) is Rayleigh distributed if R = \sqrt, where X \sim N(0, \sigma^2) and Y \sim N(0, \sigma^2) are independent normal random variables. This gives motivation to the use of the symbol \sigma in the above parametrization of the Rayleigh density. * The magnitude , z, of a standard complex normally distributed variable ''z'' is Rayleigh distributed. * The
chi distribution In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard norm ...
with ''v'' = 2 is equivalent to the Rayleigh Distribution with ''σ'' = 1. * If R \sim \mathrm (1), then R^2 has a
chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...
with parameter N, degrees of freedom, equal to two (''N'' = 2) :: =R^2\sim \chi^2(N)\ . * If R \sim \mathrm(\sigma), then \sum_^N R_i^2 has a
gamma distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...
with parameters N and \frac :: \left =\sum_^N R_i^2\right\sim \Gamma(N,\frac) . * The
Rice distribution Rice is the seed of the grass species ''Oryza sativa'' (Asian rice) or less commonly ''Oryza glaberrima'' (African rice). The name wild rice is usually used for species of the genera ''Zizania'' and ''Porteresia'', both wild and domesticated, ...
is a noncentral generalization of the Rayleigh distribution: \mathrm(\sigma) = \mathrm(0,\sigma) . * The
Weibull distribution In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice Ren ...
with the
shape parameter In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. ...
''k''=2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter \sigma is related to the Weibull scale parameter according to \lambda = \sigma \sqrt . * The
Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and used ...
describes the magnitude of a normal vector in three dimensions. * If X has an
exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
X \sim \mathrm(\lambda), then Y=\sqrt \sim \mathrm(1/\sqrt) . * The
half-normal distribution In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the hal ...
is the univariate special case of the Rayleigh distribution.


Applications

An application of the estimation of σ can be found in
magnetic resonance imaging Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio wave ...
(MRI). As MRI images are recorded as
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
images but most often viewed as magnitude images, the background data is Rayleigh distributed. Hence, the above formula can be used to estimate the noise variance in an MRI image from background data. The Rayleigh distribution was also employed in the field of
nutrition Nutrition is the biochemical and physiological process by which an organism uses food to support its life. It provides organisms with nutrients, which can be metabolized to create energy and chemical structures. Failure to obtain sufficient n ...
for linking
dietary In nutrition, diet is the sum of food consumed by a person or other organism. The word diet often implies the use of specific intake of nutrition for health or weight-management reasons (with the two often being related). Although humans are o ...
nutrient A nutrient is a substance used by an organism to survive, grow, and reproduce. The requirement for dietary nutrient intake applies to animals, plants, fungi, and protists. Nutrients can be incorporated into cells for metabolic purposes or excret ...
levels and
human Humans (''Homo sapiens'') are the most abundant and widespread species of primate, characterized by bipedalism and exceptional cognitive skills due to a large and complex brain. This has enabled the development of advanced tools, culture, ...
and
animal Animals are multicellular, eukaryotic organisms in the Kingdom (biology), biological kingdom Animalia. With few exceptions, animals Heterotroph, consume organic material, Cellular respiration#Aerobic respiration, breathe oxygen, are Motilit ...
responses. In this way, the
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
σ may be used to calculate nutrient response relationship. In the field of
ballistics Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially ranged weapon munitions such as bullets, unguided bombs, rockets or the like; the science or art of designing and a ...
, the Rayleigh distribution is used for calculating the
circular error probable In the military science of ballistics, circular error probable (CEP) (also circular error probability or circle of equal probability) is a measure of a weapon system's precision. It is defined as the radius of a circle, centered on the mean, wh ...
—a measure of a weapon's precision. In
physical oceanography Physical oceanography is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters. Physical oceanography is one of several sub-domains into which oceanography is divi ...
, the distribution of
significant wave height In physical oceanography, the significant wave height (SWH, HTSGW or ''H''s) is defined traditionally as the mean ''wave height'' ( trough to crest) of the highest third of the waves (''H''1/3). Nowadays it is usually defined as four times the ...
approximately follows a Rayleigh distribution.


See also

*
Circular error probable In the military science of ballistics, circular error probable (CEP) (also circular error probability or circle of equal probability) is a measure of a weapon system's precision. It is defined as the radius of a circle, centered on the mean, wh ...
*
Rayleigh fading Rayleigh fading is a statistical model for the effect of a propagation environment on a radio signal, such as that used by wireless devices. Rayleigh fading models assume that the magnitude of a signal that has passed through such a transmission m ...
* Rayleigh mixture distribution *
Rice distribution Rice is the seed of the grass species ''Oryza sativa'' (Asian rice) or less commonly ''Oryza glaberrima'' (African rice). The name wild rice is usually used for species of the genera ''Zizania'' and ''Porteresia'', both wild and domesticated, ...


References

{{DEFAULTSORT:Rayleigh Distribution Continuous distributions Exponential family distributions