﻿ Rayleigh distribution

# 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 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.

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