Rayleigh Distribution Calculator

The Rayleigh distribution is a continuous probability distribution named after the English physicist Lord Rayleigh. It commonly arises when a two-dimensional vector has normally distributed, independent, and identically distributed components — its magnitude then follows a Rayleigh distribution. It is widely used in signal processing, wind speed modeling, and wireless communications.

The probability density function is f(x, σ) = (x / σ²) · exp(−x² / (2σ²)) for x ≥ 0. The cumulative distribution function is F(x, σ) = 1 − exp(−x² / (2σ²)). Both depend on the scale parameter σ, which controls the spread of the distribution.

The scale parameter σ determines the shape and spread of the distribution. A larger σ shifts the distribution to the right, producing higher mean and variance values. It is not the standard deviation of the distribution itself — the actual standard deviation is derived from σ using the formula √((4 − π) / 2) · σ.

The mean of the Rayleigh distribution is σ · √(π / 2), and the mode (the peak of the PDF) equals σ. The variance is σ² · (4 − π) / 2, and the standard deviation is the square root of the variance. These measures all scale directly with the parameter σ.

The survival function, also called the complementary CDF, is S(x) = 1 − F(x) = exp(−x² / (2σ²)). It represents the probability that the random variable exceeds the value x. In reliability engineering, this is known as the reliability function.

The Rayleigh distribution has many practical uses: modeling wind speed distributions in meteorology, describing signal envelope amplitudes in wireless communications (Rayleigh fading), analyzing wave heights in oceanography, and modeling lifetime data in reliability and survival analysis.

No. The Rayleigh distribution is only defined for x ≥ 0. At x = 0, the PDF equals 0. For negative values of x, the distribution has no probability mass — f(x) = 0. The calculator automatically handles this by requiring x to be non-negative.

The Rayleigh distribution is a special case of the Weibull distribution with shape parameter k = 2. It is also related to the chi distribution with 2 degrees of freedom. If X and Y are independent standard normal variables, then √(X² + Y²) follows a Rayleigh distribution with σ = 1.