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Parameters scale: σ > 0 {displaystyle sigma >0,} Support x ∈ [ 0 , + ∞ ) {displaystyle xin [0,+infty )} PDF x σ 2 e − x 2 / ( 2 σ 2 ) {displaystyle {rac {x}{sigma ^{2}}}e^{-x^{2}/left(2sigma ^{2}ight)}} CDF 1 − e − x 2 / ( 2 σ 2 ) {displaystyle 1-e^{-x^{2}/left(2sigma ^{2}ight)}} Quantile Q ( F ; σ ) = σ − ln [ ( 1 − F ) 2 ] {displaystyle Q(F;sigma )=sigma {sqrt {-ln[(1-F)^{2}]}}} Mean σ π 2 {displaystyle sigma {sqrt {rac {pi }{2}}}} |
In probability theory and statistics, the Rayleigh distribution /ˈreɪli/ is a continuous probability distribution for positive-valued random variables.
Contents
- Definition
- Relation to random vector length
- Properties
- Differential entropy
- Differential equation
- Parameter estimation
- Confidence intervals
- Generating random variates
- Related distributions
- Applications
- References
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 into its orthogonal 2-dimensional vector components. 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.
The distribution is named after Lord Rayleigh
Definition
The probability density function of the Rayleigh distribution is
where
for
Relation to random vector length
Consider the two-dimensional vector
Let
By transforming to the polar coordinate system one has
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.
Properties
The raw moments are given by:
where
The mean and variance of a Rayleigh random variable may be expressed as:
and
The mode is
The skewness is given by:
The excess kurtosis is given by:
The characteristic function is given by:
where
where
Differential entropy
The differential entropy is given by
where
Differential equation
The pdf of the Rayleigh distribution is a solution of the following differential equation:
Parameter estimation
Given a sample of N independent and identically distributed Rayleigh random variables
Confidence intervals
To find the (1 − α) confidence interval, first find the two numbers
then
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
Related distributions
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.