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In
probability theory Probability theory or probability calculus 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 expre ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, the Rayleigh distribution is a
continuous probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
for nonnegative-valued
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s. Up to rescaling, it coincides with the
chi distribution In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with two
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
. The distribution is named after Lord Rayleigh (). A Rayleigh distribution is often observed when the overall magnitude of a vector in the plane is related to its directional
components Component may refer to: In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis * Lumped e ...
. One example where the Rayleigh distribution naturally arises is when
wind Wind is the natural movement of atmosphere of Earth, air or other gases relative to a planetary surface, planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heatin ...
velocity is analyzed in two dimensions. 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, ther ...
,
normally distributed In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is f(x ...
with equal
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
, and zero
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
, which is infrequent, then the overall wind speed (
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
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 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), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
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 ...
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. Ever ...
is :F(x;\sigma) = 1 - e^ for x \in bivariate normally distributed, centered at zero, with equal variances \sigma^2, 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
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 line, r ...
in polar coordinate system">polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
, 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 or correlations ( Hoyt distribution), or when the vector ''Y'' follows a bivariate Student ''t''-distribution (see also: Hotelling's T-squared distribution). Suppose Y is a random vector with components u,v that follows a multivariate t-distribution. 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. Ever ...
(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 specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
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), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
(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 Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
. The
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
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 of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
of a Rayleigh random variable is: :\operatorname(X) = \sqrt \sigma \approx 0.655\ \sigma The
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
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 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 ...
is given by: :\gamma_1 = \frac \approx 0.631 The excess
kurtosis In probability theory and statistics, kurtosis (from , ''kyrtos'' or ''kurtos'', meaning "curved, arching") refers to the degree of “tailedness” in the probability distribution of a real-valued random variable. Similar to skewness, kurtos ...
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 function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integrat ...
. The moment generating function 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 function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integrat ...
.


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 called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...
.


Parameter estimation

Given a sample of ''N''
independent and identically distributed Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...
Rayleigh random variables x_i with parameter \sigma, : \widehat = \!\,\frac\sum_^N x_i^2 is the
maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
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 inaccurate, closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individ ...
. :\widehat\approx \sqrt is a biased estimator that can be corrected via the formula :\sigma = \widehat \frac = \widehat \frac = \frac, where c4 is the correction factor used to unbias estimates of standard deviation for normal random variables.


Confidence intervals

To find the (1 − ''α'') confidence interval, first find the bounds ,b/math> where: :  P\left(\chi_^2 \leq a\right) = \alpha/2, \quad P\left(\chi_^2 \leq b\right) = 1 - \alpha/2 then the scale parameter will fall within the bounds :  \frac \leq \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-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 over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with ''v'' = 2 is equivalent to the Rayleigh Distribution with ''σ'' = 1: R(\sigma) \sim \sigma\chi_2^\ . * If R \sim \mathrm (1), then R^2 has a
chi-squared distribution In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with 2 degrees of freedom: =R(\sigma)^2\sim \sigma^2\chi_2^2\ . * 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 versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the g ...
with integer scale parameter N and rate parameter \frac *: \left =\sum_^N R_i^2\right\sim \Gamma\left(N,\frac\right) with integer shape parameter ''N'' and rate parameter \frac. *: \left =\sum_^N R_i^2\right\sim \Gamma\left(N,2\sigma^2\right) with integer shape parameter ''N'' and scale parameter 2\sigma^2. * The Rice distribution is a noncentral generalization of the Rayleigh distribution: \mathrm(\sigma) = \mathrm(0,\sigma) . * The Weibull distribution 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. th ...
''k'' = 2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter \sigma is related to the Weibull scale parameter according to \lambda = \sigma \sqrt . * If X has an
exponential distribution In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuousl ...
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 ha ...
is the one-dimensional equivalent of the Rayleigh distribution. * 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 use ...
is the three-dimensional equivalent 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 generate pictures of the anatomy and the physiological processes inside the body. MRI scanners use strong magnetic fields, magnetic field gradients, and ...
(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 biochemistry, biochemical and physiology, physiological process by which an organism uses food and water to support its life. The intake of these substances provides organisms with nutrients (divided into Macronutrient, macro- ...
for linking dietary
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'') or modern humans are the most common and widespread species of primate, and the last surviving species of the genus ''Homo''. They are Hominidae, great apes characterized by their Prehistory of nakedness and clothing ...
and
animal Animals are multicellular, eukaryotic organisms in the Biology, biological Kingdom (biology), kingdom Animalia (). With few exceptions, animals heterotroph, consume organic material, Cellular respiration#Aerobic respiration, breathe oxygen, ...
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 weapon munitions such as bullets, unguided bombs, rockets and the like; the science or art of designing and acceler ...
, the Rayleigh distribution is used for calculating the
circular error probable Circular error probable (CEP),Circular Error Probable (CEP), Air Force Operational Test and Evaluation Center Technical Paper 6, Ver 2, July 1987, p. 1 also circular error probability or circle of equal probability, is a measure of a weapon s ...
—a measure of a gun'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 div ...
, 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 (physics), trough to crest (physics), crest) of the highest third of the ocean surface wave, waves (''H''1/ ...
approximately follows a Rayleigh distribution.


See also

*
Circular error probable Circular error probable (CEP),Circular Error Probable (CEP), Air Force Operational Test and Evaluation Center Technical Paper 6, Ver 2, July 1987, p. 1 also circular error probability or circle of equal probability, is a measure of a weapon s ...
*
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 ...
* Rayleigh mixture distribution * Rice distribution


References

{{DEFAULTSORT:Rayleigh Distribution Continuous distributions Exponential family distributions