Rayleigh Distribution Calculator

The Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. It is uniquely defined by a single scale parameter, often denoted as sigma (σ). The mathematical core of the distribution relies on the scale value to determine its spread and peak. The probability density function (PDF) is expressed as f(x; σ) = (x / σ^2) * exp(-x^2 / (2 * σ^2)) for x ≥ 0. From this foundational equation, the cumulative distribution function (CDF), mean, variance, median, mode, and skewness are derived mathematically. For instance, the mean is calculated as σ * sqrt(π / 2), while the mode simply equals the scale parameter σ.

Consider a wind energy analysis where the scale parameter (σ) of the Rayleigh distribution is set to 4 meters per second. We want to find the Probability Density Function (PDF) and Cumulative Distribution Function (CDF) at a specific wind speed x = 3 meters per second. First, compute the PDF using the formula: f(3) = (3 / 4^2) * exp(-3^2 / (2 * 4^2)) = (3 / 16) * exp(-9 / 32) = 0.1875 * exp(-0.28125) ≈ 0.1419. Next, compute the CDF representing the probability that wind speed is less than or equal to 3 m/s: F(3) = 1 - exp(-3^2 / (2 * 4^2)) = 1 - exp(-0.28125) ≈ 0.2452 or 24.52 percent. This step-by-step logic allows you to instantly interpret probabilistic risks and performance thresholds in engineering applications.

When working with Rayleigh distributions, always verify that your input data represents nonnegative physical phenomena, as negative values violate the core assumption of the model. Ensure your scale parameter and evaluation points use consistent units of measurement to avoid scaling errors. Additionally, use the quantile function features to reverse-engineer threshold values corresponding to specific confidence intervals or safety tolerances in your project.