Rayleigh Distribution Calculator

Calculate probability density function, cumulative distribution function, and distribution properties for the Rayleigh distribution.

Last updated: March 2026

Distribution Parameters

Scale Parameter (sigma)

Value (x greater than or equal to 0)

PDF f(x)

0.243489

CDF F(x)

0.675348

Distribution Properties

Mean2.5066

Median2.3548

Mode2.0000

Variance1.7168

Std Dev1.3103

P(X greater than x)0.324652

What is Rayleigh Distribution?

The Rayleigh distribution models the magnitude of a two-dimensional vector with normally distributed components that have equal variance and zero mean. It is always non-negative and skewed to the right.

Common applications include wind speed magnitudes, radar signal processing, particle physics, and amplitude of complex signals. The distribution has a single parameter, sigma (scale parameter), which determines its shape and spread.

For a Rayleigh distribution, the mean is sigma times the square root of pi over 2 (approximately 1.253 times sigma), and the variance is (4 minus pi) divided by 2 times sigma squared.

Distribution Formulas

Key Formulas

PDF: f(x) equals (x / sigma squared) times e to the power (negative x squared divided by 2 sigma squared)

CDF: F(x) equals 1 minus e to the power (negative x squared divided by 2 sigma squared)

Mean: mu equals sigma times square root (pi / 2)

Median: equals sigma times square root (2 ln(2))

Mode: equals sigma

Variance: sigma squared equals ((4 minus pi) / 2) times sigma squared

Properties

• The distribution is always non-negative (x greater than or equal to 0)
• It is right-skewed for all values of sigma
• The mode is at x equals sigma
• The mean is always greater than the mode (1.253 times sigma)

Example: Wind Speed Magnitude

Modeling wind speed magnitude with sigma equals 2 m/s at x equals 3 m/s:

PDF f(3): equals (3 / 4) times e to the power (negative 9/8)

approximately 0.232 per m/s

CDF F(3): equals 1 minus e to the power (negative 9/8)

approximately 0.676

FAQ About Rayleigh

When should I use Rayleigh distribution?

Use it for modeling magnitudes of two-dimensional vectors with normal components, wind speeds, radar signals, or any positive skewed data representing amplitudes.

What does sigma parameter control?

Sigma is the scale parameter. Larger sigma values shift and spread the distribution. It determines the mean, mode, and variance of the distribution.

How does CDF relate to PDF?

CDF is the integral of PDF. It represents the cumulative probability that a value is less than or equal to x.

Why is the mode different from the mean?

The Rayleigh distribution is right-skewed. The mode is at sigma, but the mean is at 1.253 times sigma due to the right tail.

Can I have negative x values?

No. The Rayleigh distribution is only defined for x greater than or equal to 0, as it models magnitudes.

What is the relationship to normal distribution?

If X and Y are independent normal random variables with mean 0 and the same sigma, then R equals square root (X squared plus Y squared) follows Rayleigh distribution.