Rayleigh Distribution
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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, the Rayleigh distribution is a continuous probability distribution 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 p ...
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 no ...
with two
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
. 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. 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 ...
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, there ...
,
normally distributed In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is ...
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 number ...
, 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 '' ari ...
, 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 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) c ...
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 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
double integral 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 t ...
, 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. 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 t ...
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) c ...
(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 th ...
. 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 '' ari ...
of a Rayleigh random variable is thus : :\mu(X) = \sigma \sqrt\ \approx 1.253\ \sigma. The standard deviation 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 number ...
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 unimo ...
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, kur ...
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 point ...
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- elementa ...
. 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 complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non- elementa ...
.


Differential entropy

The differential entropy is given by :H = 1 + \ln\left(\frac \sigma \right) + \frac \gamma 2 where \gamma is the Euler–Mascheroni constant.


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 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 sta ...
estimate and also is unbiased. :\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 Uniform distribution may refer to: * Continuous uniform distribution * Discrete uniform distribution * Uniform distribution (ecology) * Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
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. 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 no ...
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-squar ...
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 dis ...
with parameters N and \frac :: \left =\sum_^N R_i^2\right\sim \Gamma(N,\frac) . * The Rice distribution 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 R ...
with the shape parameter ''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 use ...
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 averag ...
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 univariate special case of the Rayleigh distribution.


Applications

An application of the estimation of σ can be found in magnetic resonance imaging (MRI). As MRI images are recorded as complex 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 sufficien ...
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 excr ...
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 ...
, 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, ...
—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 di ...
, the distribution of significant wave height 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, ...
*
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 transmissio ...
* Rayleigh mixture distribution * Rice distribution


References

{{DEFAULTSORT:Rayleigh Distribution Continuous distributions Exponential family distributions