An Introduction to the Rayleigh Distribution

The Rayleigh distribution is a continuous probability distribution used to model random variables that can only take on values equal to or greater than zero.

It has the following probability density function:

f(x; σ) = (x/σ2)e-x2/(2σ2)

where σ is the scale parameter of the distribution.

Properties of the Rayleigh Distribution

The Rayleigh Distribution has the following properties:

  • Mean: σ√π/2
  • Variance: ((4-π)/2)σ2
  • Mode: σ

Since π has a known numerical value, we can simplify the properties as follows:

  • Mean: 1.253σ
  • Variance: 0.429σ2
  • Mode: σ

Visualizing the Rayleigh Distribution

The following chart shows the shape of the Rayleigh distribution when it takes on different values for the scale parameter:

Rayleigh probability distribution

Note that the larger the value for the scale parameter σ, the wider the distribution becomes.

Bonus: For those who are curious, we used the following R code to generate the chart above:

#load VGAM package

#create density plots
curve(drayleigh(x, scale = 0.5), from=0, to=10, col='green')
curve(drayleigh(x, scale = 1), from=0, to=10, col='red', add=TRUE)
curve(drayleigh(x, scale = 2), from=0, to=10, col='blue', add=TRUE)
curve(drayleigh(x, scale = 4), from=0, to=10, col='purple', add=TRUE)

#add legend
legend(6, 1, legend=c("σ=0.5", "σ=1", "σ=2", "σ=4"),
       col=c("green", "red", "blue", "purple"), lty=1, cex=1.2)

Relation to Other Distributions

The Rayleigh distribution has the following relationship with other probability distributions:

1. When the scale parameter (σ) is equal to 1, the Rayleigh distribution is equal to a Chi-Square distribution with 2 degrees of freedom.

2. The Rayleigh distribution is a special case of the Weibull distribution with a shape parameter of k = 2.

3. The Rayleigh distribution with scale parameter σ is equal to the Rice distribution with Rice(0, σ).


In practice, the Rayleigh distribution is used in a variety of applications including:

1. The Rayleigh distribution is used to model wave behavior in the ocean, including the time it takes waves to crest and the max height reached by waves.

2. The Rayleigh distribution is used to model the behavior of background data in magnetic resonance imaging, more commonly known as MRI.

3. The Rayleigh distribution is used in the field of nutrition to model the relationship between nutrient levels and nutrient response in both humans and animals.

Additional Resources

An Introduction to the Normal Distribution
An Introduction to the Binomial Distribution
An Introduction to the Poisson Distribution

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