Probability density function
Cumulative distribution function
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. It is essentially a chi distribution with two degrees of freedom.
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 velocity is analyzed in two dimensions. Assuming that each component is uncorrelated, normally distributed with equal variance, and zero mean, then the overall wind speed (vector 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.
Relation to random vector length
Consider the two-dimensional vector which has components that are normally distributed, centered at zero, and independent. Then and have density functions
Let be the length of . That is, Then has cumulative distribution function
where is the disk
Finally, the probability density function for is the derivative of its cumulative distribution function, which by the fundamental theorem of calculus is
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, or when the vector Y follows a bivariate Student t-distribution.
Generalization to bivariate Student's-t distribution
Suppose is a random vector with components 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:
Let be the magnitude of . Then the cumulative distribution function (CDF) of the magnitude is:
where is the disk defined by:
Converting to polar coordinates leads to the CDF becoming:
Finally, the probability density function (PDF) of the magnitude may be derived:
In the limit as , the Rayleigh distribution is recovered because:
The raw moments are given by:
where is the gamma function.
The mean of a Rayleigh random variable is thus :
The standard deviation of a Rayleigh random variable is:
The variance of a Rayleigh random variable is :
The mode is and the maximum pdf is
The skewness is given by:
The excess kurtosis is given by:
The characteristic function is given by:
where is the error function.
where is the Euler–Mascheroni constant.
Given a sample of N independent and identically distributed Rayleigh random variables with parameter ,
- is a biased estimator that can be corrected via the formula
To find the (1 − α) confidence interval, first find the bounds where:
then the scale parameter will fall within the bounds
Generating random variates
Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate
has a Rayleigh distribution with parameter . This is obtained by applying the inverse transform sampling-method.
- is Rayleigh distributed if , where and 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 of a standard complex normally distributed variable z will have the Rayleigh distribution.
- The chi distribution with v = 2 is equivalent to the Rayleigh Distribution with σ = 1.
- If , then has a chi-squared distribution with parameter , degrees of freedom, equal to two (N = 2)
- If , then has a gamma distribution with parameters and
- The Rice distribution is a noncentral generalization of the Rayleigh distribution: .
- The Weibull distribution with the "shape parameter" k=2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter is related to the Weibull scale parameter according to
- The Maxwell–Boltzmann distribution describes the magnitude of a normal vector in three dimensions.
- If has an exponential distribution , then
- The half-normal distribution is the univariate special case of the Rayleigh distribution.
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 for linking dietary nutrient levels and human and animal responses. In this way, the parameter σ may be used to calculate nutrient response relationship.
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