Rayleigh Distribution Calculator
Calculate Rayleigh distribution PDF, CDF, and P(X > x) with scale parameter σ. Free online statistics tool for mean, median, mode, variance, and interactive PDF curve chart.
About This Calculator
The Rayleigh Distribution Calculator computes probabilities and statistics for the Rayleigh distribution — a continuous probability distribution named after Lord Rayleigh that models the magnitude of a two-dimensional vector whose components are independent normal random variables with zero mean and equal variance. Given the scale parameter σ (sigma) and a target value x, the calculator returns the probability density function (PDF), cumulative distribution function (CDF), tail probability P(X > x), mean, median, mode, variance, standard deviation, and skewness. The interactive chart visualizes the full PDF curve.
The Rayleigh distribution's PDF formula is f(x; σ) = (x/σ²)e^(-x²/(2σ²)) for x ≥ 0. The CDF is F(x; σ) = 1 - e^(-x²/(2σ²)), and the survival function P(X > x) = e^(-x²/(2σ²)). The mean equals σ√(π/2), median equals σ√(2·ln(2)), mode equals σ, and variance equals σ²(4-π)/2. A key property is its linearly increasing hazard rate, making it suitable for modeling wear-out failures, unlike the exponential distribution with its constant hazard rate.
The Rayleigh distribution is a special case of the Weibull distribution (with shape k = 2) and is intimately related to the normal distribution: if X and Y are independent N(0, σ²), then √(X² + Y²) follows a Rayleigh distribution. It is widely used in reliability engineering, telecommunications (modeling signal fading), physics (sound and light radiation), weather forecasting (wind speed), and medical imaging (MRI noise).
Regional Notes
The Rayleigh distribution is a universal mathematical concept used across all regions. In the United States, it is taught in AP Statistics and college-level probability and engineering courses. In India, it features in the curriculum for engineering, statistics, and physics programs, particularly in the context of reliability theory and signal processing. In the United Kingdom, it appears in A-level Further Statistics and undergraduate mathematics, physics, and engineering courses. The mathematical formulas and interpretation are identical worldwide.
Frequently Asked Questions
What is the Rayleigh distribution?
The Rayleigh distribution is a continuous probability distribution named after Lord Rayleigh. It arises when you take the magnitude of two independent normal random variables with zero mean and equal variance. It is a special case of the Weibull distribution and is closely related to the exponential distribution — the square root of an exponential random variable follows a Rayleigh distribution.
How do you calculate the PDF of the Rayleigh distribution?
The probability density function (PDF) of the Rayleigh distribution is f(x; σ) = (x/σ²)e^(-x²/(2σ²)) for x ≥ 0, where σ is the scale parameter. The PDF starts at f(0) = 0, rises to a maximum at the mode x = σ, and then decays exponentially. The shape is right-skewed and differs from the exponential distribution which has its maximum at x = 0.
What is the CDF formula for the Rayleigh distribution?
The cumulative distribution function (CDF) of the Rayleigh distribution is P(X ≤ x) = 1 - e^(-x²/(2σ²)). The survival function or tail probability is P(X > x) = e^(-x²/(2σ²)). The quantile function (inverse CDF) is Q(p) = σ√(-2ln(1-p)).
What are the mean, median, mode, and variance of the Rayleigh distribution?
The mean is σ√(π/2), the median is σ√(2·ln(2)) ≈ σ√(ln(4)), and the mode is σ. The variance is σ²(4-π)/2 and the standard deviation is σ√((4-π)/2). The skewness is approximately 0.631 — a constant that does not depend on σ. Unlike the exponential distribution where mean equals standard deviation, the Rayleigh mean is larger than its standard deviation.
What is the relationship between Rayleigh and normal distributions?
If X and Y are independent normal random variables with zero mean and equal standard deviation, then the magnitude R = √(X² + Y²) follows a Rayleigh distribution. This relationship makes the Rayleigh distribution fundamental in physics for modeling the magnitude of a two-dimensional vector whose components are normally distributed, such as wind speed from orthogonal wind components.
Where is the Rayleigh distribution used in real-world applications?
The Rayleigh distribution is widely used in electrical engineering to model the lifetime of capacitors and transformers, in physics for sound and light radiation, in weather forecasting for wind speed modeling, in hydrology for wave height analysis, in telecommunications for modeling signal fading in wireless channels, and in medical imaging for modeling noise in MRI magnitude images.
How do you use the Rayleigh distribution calculator?
To use the Rayleigh distribution calculator, enter the scale parameter σ (sigma) which determines the spread of the distribution. Then enter the value x for which you want to compute probabilities. The calculator instantly returns PDF, CDF, P(X > x), mean, median, mode, variance, standard deviation, and skewness. The interactive chart shows the full PDF curve across the distribution's domain.
Is the Rayleigh distribution related to the Weibull distribution?
Yes, the Rayleigh distribution is a special case of the Weibull distribution with shape parameter k = 2 and scale parameter λ = σ√2. The exponential distribution is also a special case of the Weibull distribution (with k = 1). While the exponential distribution has a constant hazard rate, the Rayleigh distribution has a linearly increasing hazard rate, making it suitable for modeling aging and wear-out failures.