34  Rayleigh Distribution

The Rayleigh distribution describes the magnitude (length) of a two-dimensional vector whose components are independent zero-mean Gaussian random variables. It is the standard model for signal envelope in wireless communications and for wind speed distributions.

Formally, the random variate \(X\) defined for the range \(X \in [0, \infty)\), is said to have a Rayleigh Distribution (i.e. \(X \sim \text{Rayleigh}(\sigma)\)) with scale parameter \(\sigma > 0\). If \(X_1, X_2 \overset{\text{i.i.d.}}{\sim} N(0, \sigma^2)\) then \(\sqrt{X_1^2 + X_2^2} \sim \text{Rayleigh}(\sigma)\).

34.1 Probability Density Function

\[ f(x) = \frac{x}{\sigma^2}\exp\!\left(-\frac{x^2}{2\sigma^2}\right), \quad x \geq 0 \]

The figure below shows examples of the Rayleigh Probability Density Function for different scale values.

Code 
drayleigh <- function(x, sigma) {
  ifelse(x >= 0, x / sigma^2 * exp(-x^2 / (2 * sigma^2)), 0)
}

par(mfrow = c(2, 2))
x <- seq(0, 12, length = 500)

plot(x, drayleigh(x, 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(sigma == 1))

plot(x, drayleigh(x, 2), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(sigma == 2))

plot(x, drayleigh(x, 3), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(sigma == 3))

plot(x, drayleigh(x, 5), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(sigma == 5))

par(mfrow = c(1, 1))
Figure 34.1: Rayleigh Probability Density Function for various scale values

34.2 Purpose

The Rayleigh distribution models the Euclidean magnitude of a 2D random vector with independent, equal-variance Gaussian components. Its elegant mathematical form and connection to the Normal distribution make it ideal for modeling magnitude quantities in physics, engineering, and communications. Common applications include:

  • Wireless communications: envelope of a narrowband signal with Rayleigh fading
  • Wind energy: marginal distribution of wind speed for power generation modeling
  • Underwater acoustics: noise envelope in sonar systems
  • Optics: speckle patterns in laser imaging
  • Navigation: radial positioning error from two independent Gaussian components

Relation to the discrete setting. The Rayleigh distribution models continuous 2D Euclidean magnitude; the closest discrete analog is a 2D random walk return distance. Conceptually related to the Geometric distribution as both model “distance” to first event in their respective spaces.

34.3 Distribution Function

\[ F(x) = 1 - \exp\!\left(-\frac{x^2}{2\sigma^2}\right), \quad x \geq 0 \]

The figure below shows the Rayleigh Distribution Function for \(\sigma = 3\).

Code 
prayleigh <- function(x, sigma) {
  ifelse(x >= 0, 1 - exp(-x^2 / (2 * sigma^2)), 0)
}

x <- seq(0, 12, length = 500)
plot(x, prayleigh(x, 3), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "F(x)", main = "Rayleigh Distribution Function",
     sub = expression(sigma == 3))
Figure 34.2: Rayleigh Distribution Function (sigma = 3)

34.4 Moment Generating Function

The moment generating function has no simple closed form. Raw moments are computed directly from the density:

\[ \mu_n' = \sigma^n\, 2^{n/2}\,\Gamma\!\left(1 + \frac{n}{2}\right) \]

34.5 1st Uncentered Moment

\[ \mu_1' = \sigma\sqrt{\frac{\pi}{2}} \]

34.6 2nd Uncentered Moment

\[ \mu_2' = 2\sigma^2 \]

34.7 3rd Uncentered Moment

\[ \mu_3' = \frac{3\sigma^3\sqrt{2\pi}}{2} \]

34.8 4th Uncentered Moment

\[ \mu_4' = 8\sigma^4 \]

34.9 2nd Centered Moment

\[ \mu_2 = \frac{(4-\pi)\sigma^2}{2} \]

34.10 3rd Centered Moment

\[ \mu_3 = \mu_3' - 3\mu_2'\mu_1' + 2(\mu_1')^3 \]

34.11 4th Centered Moment

\[ \mu_4 = \mu_4' - 4\mu_3'\mu_1' + 6\mu_2'(\mu_1')^2 - 3(\mu_1')^4 \]

34.12 Expected Value

\[ \text{E}(X) = \sigma\sqrt{\frac{\pi}{2}} \]

34.13 Variance

\[ \text{V}(X) = \frac{(4-\pi)\sigma^2}{2} \]

34.14 Median

\[ \text{Med}(X) = \sigma\sqrt{2\ln 2} \]

34.15 Mode

\[ \text{Mo}(X) = \sigma \]

The density has a unique maximum at \(x = \sigma\).

34.16 Coefficient of Skewness

\[ g_1 = \frac{2\sqrt{\pi}(\pi - 3)}{(4 - \pi)^{3/2}} \approx 0.631 \]

This is a fixed constant, independent of \(\sigma\).

34.17 Coefficient of Kurtosis

\[ g_2 = \frac{32 - 3\pi^2}{(4-\pi)^2} \approx 3.245089 \]

This is a fixed constant, independent of \(\sigma\). The corresponding excess kurtosis is \(g_2 - 3 \approx 0.245089\).

34.18 Parameter Estimation

The MLE of \(\sigma^2\) has an exact closed form:

\[ \hat\sigma^2 = \frac{1}{2n}\sum_{i=1}^n x_i^2 \]

set.seed(42)
sigma_true <- 5

# Simulate Rayleigh data via 2D Gaussian magnitude
x1 <- rnorm(100, 0, sigma_true)
x2 <- rnorm(100, 0, sigma_true)
x_obs <- sqrt(x1^2 + x2^2)

# MLE
sigma_hat <- sqrt(mean(x_obs^2) / 2)
cat("MLE sigma:", round(sigma_hat, 4), "\n")
cat("True sigma:", sigma_true, "\n")
MLE sigma: 4.8627 
True sigma: 5

34.19 R Module

34.19.1 RFC

The Rayleigh Distribution module is available in RFC under the menu “Distributions / Rayleigh Distribution”.

34.19.3 R Code

The following code demonstrates Rayleigh probability calculations:

sigma <- 5

drayleigh <- function(x, sigma) ifelse(x >= 0, x/sigma^2 * exp(-x^2/(2*sigma^2)), 0)
prayleigh <- function(x, sigma) ifelse(x >= 0, 1 - exp(-x^2/(2*sigma^2)), 0)
qrayleigh <- function(p, sigma) sigma * sqrt(-2 * log(1 - p))

# Density at x = 5
drayleigh(5, sigma)

# P(X <= 10)
prayleigh(10, sigma)

# P(gust > 10)
1 - prayleigh(10, sigma)

# Mean and mode
cat("Mean:", sigma * sqrt(pi / 2), "\n")
cat("Mode:", sigma, "\n")
[1] 0.1213061
[1] 0.8646647
[1] 0.1353353
Mean: 6.266571 
Mode: 5

34.20 Example

Wind speeds at a coastal site are modeled as \(X \sim \text{Rayleigh}(\sigma = 5)\) m/s. The mean wind speed is \(\sigma\sqrt{\pi/2} \approx 6.27\) m/s. We compute the probability of a gust exceeding 10 m/s.

sigma <- 5
prayleigh <- function(x, sigma) ifelse(x >= 0, 1 - exp(-x^2/(2*sigma^2)), 0)

# P(gust > 10 m/s)
cat("P(gust > 10 m/s):", 1 - prayleigh(10, sigma), "\n")

# Mean wind speed
cat("Mean wind speed (m/s):", sigma * sqrt(pi / 2), "\n")

# Median wind speed
cat("Median wind speed (m/s):", sigma * sqrt(2 * log(2)), "\n")
P(gust > 10 m/s): 0.1353353 
Mean wind speed (m/s): 6.266571 
Median wind speed (m/s): 5.88705
Interactive Shiny app (click to load).
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34.21 Random Number Generator

Rayleigh random variates are generated via the inverse-CDF method. Since \(F(x) = 1 - \exp(-x^2/(2\sigma^2))\), solving for \(X\) gives:

\[ X = \sigma\sqrt{-2\ln U} \sim \text{Rayleigh}(\sigma) \quad \text{when } U \sim \text{U}(0,1) \]

set.seed(123)
n <- 1000; sigma <- 5

# Inverse-transform method
u <- runif(n)
x_inv <- sigma * sqrt(-2 * log(u))

cat("Simulated mean:", round(mean(x_inv), 4), "\n")
cat("Theoretical mean:", sigma * sqrt(pi / 2), "\n")
cat("Simulated var:", round(var(x_inv), 4), "\n")
cat("Theoretical var:", (4 - pi) * sigma^2 / 2, "\n")
Simulated mean: 6.2946 
Theoretical mean: 6.266571 
Simulated var: 10.7059 
Theoretical var: 10.73009
Interactive Shiny app (click to load).
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34.22 Property 1: Special Case of Weibull

The Rayleigh distribution is the Weibull distribution with shape \(k = 2\) and scale \(\lambda = \sigma\sqrt{2}\):

\[ \text{Rayleigh}(\sigma) = \text{Weibull}(2,\, \sigma\sqrt{2}) \]

See Chapter 31.

34.23 Property 2: 2D Gaussian Magnitude

If \(X_1, X_2 \overset{\text{i.i.d.}}{\sim} N(0, \sigma^2)\) then:

\[ \sqrt{X_1^2 + X_2^2} \sim \text{Rayleigh}(\sigma) \]

This makes the Rayleigh distribution the natural model for 2D positioning errors and for the envelope of two-dimensional noise.

34.24 Property 3: Constant Coefficient of Variation

The coefficient of variation (CV = standard deviation / mean) is:

\[ \text{CV} = \sqrt{\frac{4}{\pi} - 1} \approx 0.523 \]

This is a universal constant, independent of \(\sigma\).