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/σ^{2})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)σ^{2}**Mode:**σ

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

**Mean:**1.253σ**Variance:**0.429σ^{2}**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:

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 library(VGAM) #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).

**Applications**

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