Rayleigh Distribution Calculator

What This Rayleigh Distribution Calculator Does

This calculator evaluates common Rayleigh distribution functions from a scale parameter $\sigma$. You can compute the probability density function, cumulative distribution function, quantiles, interval probabilities, and summary measures such as mean and variance.

Core Rayleigh Formulas

$$f(x) = \frac{x}{\sigma^2}e^{-x^2/(2\sigma^2)}, \qquad F(x) = 1 - e^{-x^2/(2\sigma^2)}, \qquad x \ge 0$$

$$Q(p) = \sigma \sqrt{-2\ln(1-p)}$$

Common Measures

• Mean: $\sigma \sqrt{\pi/2}$
• Median: $\sigma \sqrt{2\ln 2}$
• Mode: $\sigma$
• Variance: $\sigma^2(4-\pi)/2$

Where The Rayleigh Distribution Appears

Rayleigh models show up in wind speed analysis, wave-height approximations, signal processing, and physics problems involving magnitudes built from two independent normal components. If you need a related continuous model, explore our Normal Distribution Calculator.