In
probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
, the
Rayleigh distribution (
pronounced: /'reɪlɪ/) is a
continuous probability distributionIn probability theory, a probability distribution is called continuous if its cumulative distribution function is continuous . This is equivalent to saying that for random variables X with the distribution in question, Pr[X = a] = 0 for all real numbers a, i.e.: the probability that X attains the...
. It can arise when a two-dimensional vector (e.g.
windWind is the flow of air or other gases that compose an atmosphere . On Earth, wind consists of the bulk movement of air...
velocityIn physics, velocity is the rate of change of position. It is a vector physical quantity; both speed and direction are required to define it. In the SI system, it is measured in meters per second: or ms-1. The scalar absolute value of velocity is speed...
data, as measured with an anemometer and wind vane, which consists of a speed value and a direction) has elements that are
normally distributedIn probability theory and statistics, the normal distribution or Gaussian distribution is a continuous probability distribution that describes data that cluster around a mean or average. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known...
, are
uncorrelatedIn probability theory and statistics, two real-valued random variables are said to be uncorrelated if their covariance is zero.Uncorrelated random variables have a correlation coefficient of zero, except in the trivial case when both variables have variance zero...
, and have equal
varianceIn probability theory and statistics, the variance of a random variable or distribution is the expected square deviation of that variable from its expected value or mean, or to put it another way: variance is the measure of the amount of variation of all the scores for a variable...
. The vector’s magnitude (e.g.
windWind is the flow of air or other gases that compose an atmosphere . On Earth, wind consists of the bulk movement of air...
speedSpeed is the rate of motion, or equivalently the rate of change of distance.Speed is a scalar quantity with dimensions length/time; the equivalent vector quantity to speed is velocity. Speed is measured in the same physical units of measurement as velocity, but does not contain the element of...
) will then have a Rayleigh distribution. The distribution can also arise in the case of random complex numbers whose real and imaginary components are
i.i.d.In probability theory and statistics, a sequence or other 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....
GaussianGAUSSIAN is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70...
.
In
probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
, the
Rayleigh distribution (
pronounced: /'reɪlɪ/) is a
continuous probability distributionIn probability theory, a probability distribution is called continuous if its cumulative distribution function is continuous . This is equivalent to saying that for random variables X with the distribution in question, Pr[X = a] = 0 for all real numbers a, i.e.: the probability that X attains the...
. It can arise when a two-dimensional vector (e.g.
windWind is the flow of air or other gases that compose an atmosphere . On Earth, wind consists of the bulk movement of air...
velocityIn physics, velocity is the rate of change of position. It is a vector physical quantity; both speed and direction are required to define it. In the SI system, it is measured in meters per second: or ms-1. The scalar absolute value of velocity is speed...
data, as measured with an anemometer and wind vane, which consists of a speed value and a direction) has elements that are
normally distributedIn probability theory and statistics, the normal distribution or Gaussian distribution is a continuous probability distribution that describes data that cluster around a mean or average. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known...
, are
uncorrelatedIn probability theory and statistics, two real-valued random variables are said to be uncorrelated if their covariance is zero.Uncorrelated random variables have a correlation coefficient of zero, except in the trivial case when both variables have variance zero...
, and have equal
varianceIn probability theory and statistics, the variance of a random variable or distribution is the expected square deviation of that variable from its expected value or mean, or to put it another way: variance is the measure of the amount of variation of all the scores for a variable...
. The vector’s magnitude (e.g.
windWind is the flow of air or other gases that compose an atmosphere . On Earth, wind consists of the bulk movement of air...
speedSpeed is the rate of motion, or equivalently the rate of change of distance.Speed is a scalar quantity with dimensions length/time; the equivalent vector quantity to speed is velocity. Speed is measured in the same physical units of measurement as velocity, but does not contain the element of...
) will then have a Rayleigh distribution. The distribution can also arise in the case of random complex numbers whose real and imaginary components are
i.i.d.In probability theory and statistics, a sequence or other 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....
GaussianGAUSSIAN is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70...
. In that case, the
absolute valueIn mathematics, the absolute value of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.The absolute value of a number is denoted by ....
of the complex number is Rayleigh-distributed. The distribution was so named after
Lord RayleighJohn William Strutt, 3rd Baron Rayleigh OM was an English physicist who, with William Ramsay, discovered the element argon, an achievement for which he earned the Nobel Prize for Physics in 1904...
.
The Rayleigh probability density function is
for
Properties
The raw
momentsThe concept of moment in mathematics evolved from the concept of moment in physics. The nth moment of a real-valued function f of a real variable about a value c is...
are given by:
where is the
Gamma functionIn mathematics, the Gamma function is an extension of the factorial function to real and complex numbers...
.
The
meanIn statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
and
varianceIn probability theory and statistics, the variance of a random variable or distribution is the expected square deviation of that variable from its expected value or mean, or to put it another way: variance is the measure of the amount of variation of all the scores for a variable...
of a Rayleigh
random variableIn mathematics, random variables are used in the study of probability. They were developed to assist in the analysis of games of chance, stochastic events, and the results of scientific experiments by capturing only the mathematical properties necessary to answer probabilistic questions...
may be expressed as:
and
The mode is and the maximum pdf is
The skewness is given by:
The excess
kurtosisIn probability theory and statistics, kurtosis is a measure of the "peakedness" of the probability distribution of a real-valued random variable...
is given by:
The
characteristic functionIn probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...
is given by:
where is the complex
error functionIn mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations...
. The moment generating function is given by
where is the
error functionIn mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations...
.
Information entropy
The
information entropyIn information theory, entropy is a measure of the uncertainty associated with a random variable. The term by itself in this context usually refers to the Shannon entropy, which quantifies, in the sense of an expected value, the information contained in a message, usually in units such as bits...
is given by
where is the
Euler–Mascheroni constantThe Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
.
Parameter estimation
Given
N independent and identically distributed Rayleigh random variables with parameter , the
maximum likelihoodMaximum likelihood estimation is a popular statistical method used for fitting a statistical model to data, and providing estimates for the model's parameters....
estimate of is
Generating Rayleigh-distributed random variates
Given a random variate
U drawn from the
uniform distribution-Probability theory:* Discrete uniform distribution* Continuous uniform distributionThey share the property that they have a bounded range, and are weakly unimodal where any members of their support can be taken to be the mode....
in the interval
(0, 1), then the variate
has a Rayleigh distribution with parameter . This follows from the form of the cumulative distribution function. Given that U is uniform, (1–U) has the same uniformity and the above may be simplified to
Note that if you are generating random numbers belonging to [0,1), exclude zero values to avoid the natural log of zero.
Related distributions
- is a Rayleigh distribution if , where and are two independent normal distribution
In probability theory and statistics, the normal distribution or Gaussian distribution is a continuous probability distribution that describes data that cluster around a mean or average. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known...
s. (This gives motivation to the use of the symbol "sigma" in the above parameterization of the Rayleigh density.)
- If , then has a chi-square distribution
In probability theory and statistics, the chi-square distribution is one of the most widely used theoretical probability distributions in inferential statistics, e.g., in statistical significance tests...
with two degrees of freedom:
- If has an exponential distribution
In probability theory and statistics, the exponential distributions are a class of continuous probability distributions. They describe the times between events in a Poisson process, i.e...
, then .
- If , then has a gamma distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. It has a scale parameter θ and a shape parameter k...
with parameters and : .
- The Chi distribution
In probability theory and statistics, the chi distribution is a continuous probability distribution. The distribution usually arises when a k-dimensional vector's orthogonal components are independent and each follow a standard normal distribution. The length of the vector will then have a chi ...
is a generalization of the Rayleigh distribution.
- The Rice distribution
In probability theory and statistics, the Rice distribution or Rician distribution, named after Stephen O. Rice, is a continuous probability distribution.-Characterization:The probability density function is...
is a generalization of the Rayleigh distribution.
- The Weibull distribution
In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Waloddi Weibull who described it in detail in 1951, although it was first identified by and first applied by to describe the size distribution of particles...
is a generalization of the Rayleigh distribution.
- The Maxwell–Boltzmann distribution
The Maxwell–Boltzmann distribution describes particle speeds in gases, where the particles do not constantly interact with each other but move freely between short collisions. It describes the probability of a particle's speed being near a given value as a function of the temperature of the...
describes the magnitude of a normal vector in three dimensions.
See also
- 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 medium will vary randomly, or fade, according to a...
- Rice distribution
In probability theory and statistics, the Rice distribution or Rician distribution, named after Stephen O. Rice, is a continuous probability distribution.-Characterization:The probability density function is...
- The SOCR
The Statistics Online Computational Resource is a suite of online tools and interactive aids for hands-on learning and teaching concepts in statistical analysis and probability theory developed at the University of California, Los Angeles...
Resource provides interactive interface to Rayleigh distribution.