Rayleigh distribution
In probability theory and statistics is called a continuous probability distribution with Rayleigh ( John William Strutt after, 3rd Baron Rayleigh ).
If the components of a two-dimensional random vector normally distributed and are statistically independent, then the amount is Rayleigh distributed. This occurs for example in a radio technology used transmission channel in mobile radio systems, if there is for example a mobile phone, no line of sight between the transmitter, such as a base station and the receiver. The by multipath propagation through various, random reflection and scattering, for example, on building walls and other obstacles that impair transmission channel can then be modeled as so-called Rayleigh channel using the Rayleigh distribution.
The distribution of 10 -minute average values of the wind speed is also frequently described by a Rayleigh distribution, if not a two-parameter Weibull distribution is to be selected.
- 4.1 Relationship to the chi -square distribution
- 4.2 Relationship to the Weibull distribution
- 4.3 Relationship to the exponential distribution
- 4.4 Relationship to the gamma distribution
- 4.5 Relationship with the normal distribution
Definition
A continuous random variable is called Rayleigh -distributed with parameters when the probability density
Possesses. Hence the distribution function
Properties
Moments
The moments of arbitrary order can be calculated using the following formula:
Where the gamma function is.
Expected value
The expected value is given by
Variance
Is the variance of the distribution
Thus, the ratio between the expected value and standard deviation of this distribution is constant:
Skew
For the skewness is obtained
Curvature ( kurtosis )
The curvature is given by
Characteristic function
The characteristic feature is
The complex error function.
Moment generating function
The moment generating function is given by
Which in turn is the error function.
Entropy
The entropy, expressed in nats, results to
The Euler - Mascheroni constant.
Mode
The maximum is reached, the Rayleigh distribution for because
For. This is the mode of the Rayleigh distribution.
At the maximum has the value
Parameter estimation
The maximum likelihood estimate of via:
Relations with other distributions
The Chi - distribution, Weibull distribution and Rice distribution are generalizations of the Rayleigh distribution.
Relationship with the chi -square distribution
Then, if the chi-square distributed with two degrees of freedom:
Relationship with the Weibull distribution
Relationship to the exponential distribution
When is exponentially distributed, then.
Relationship to the gamma distribution
If then is gamma distributed with parameters and.
Relation to the normal distribution
Rayleigh distributed when where and are two statistically independent normal distributions.