Rayleigh Distribution


< Probability distributions

The Rayleigh distribution is a continuous probability distribution named after English Lord Rayleigh. It is a special case of the Weibull distribution with a scale parameter of 2. When a Rayleigh is set with a shape parameter (σ) of 1, it is equal to a chi square distribution with 2 degrees of freedom.

The Rayleigh distribution is often used to model data that is distributed across multiple categories, such as the size of particles in a sample or the intensity of light at different wavelengths. It can also be used to model data that varies over time, such as the arrival rate of customers at a store or the amount of time it takes for an event to occur.

Rayleigh Distribution Parameters

The Rayleigh distribution is defined by two parameters: the scale parameter (σ) and the shape parameter (θ). The scale parameter determines the spread of the distribution, while the shape parameter determines its skewness.

The mean and variance of the Rayleigh distribution are both equal to σ2. The mode is equal to σ * √(ln(4)). The skewness is equal to 2/√(ln(4)), and the kurtosis is equal to 6 + ln(4).

PDF of the Rayleigh distribution for different shape parameter σ.

The notation X Rayleigh(σ) means that the random variable X has a Rayleigh distribution with shape parameter σ. The probability density function (X > 0) is:

(x/σ2)e-x2/2σ2

Applications of the Rayleigh Distribution

The Rayleigh distribution has many applications in real-world situations. For example, it can be used to model data that is distributed across multiple categories, such as the size of particles in a sample or the intensity of light at different wavelengths. It can also be used to model data that varies over time, such as the arrival rate of customers at a store or the amount of time it takes for an event to occur.

In addition, the Rayleigh distribution can be used to estimate unknown parameters in other distributions, such as the Gaussian distribution. Estimating unknown parameters is important in statistics because it allows us to make predictions about future events. For example, if we know that the mean and variance of a dataset are both equal to 5, we can use this information to predict that 95% of all future values will fall between 0 and 10.

Conclusion

The Rayleigh distribution is a continuous probability distribution named after Lord Rayleigh. It’s often used to model data that varies across multiple categories or over time. Additionally, it can be used to estimate unknown parameters in other distributions. Estimating unknown parameters is important in statistics because it allows us make predictions about future events!


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