Rayleigh Distribution Calculator - PDF, CDF & Stats
Calculate PDF, CDF, complementary CDF, mean, median, mode, and variance of the Rayleigh distribution for any scale parameter σ and value x.
Enter the scale parameter σ (must be positive) and a value x (non-negative) to see all key properties of the Rayleigh distribution instantly.
Rayleigh Distribution Calculator - PDF, CDF & Stats
Calculate PDF, CDF, complementary CDF, mean, median, mode, and variance of the Rayleigh distribution for any scale parameter σ and value x.
About the Rayleigh Distribution Calculator
The Rayleigh distribution is a continuous probability distribution for non-negative random variables, named after Lord Rayleigh who originally derived it in the context of sound wave amplitudes. It is defined entirely by a single parameter, σ (the scale parameter), which simultaneously represents the mode of the distribution — the most probable value — and governs the spread of the entire distribution.
The probability density function (PDF) is given by f(x; σ) = (x/σ²) · exp(−x²/(2σ²)) for x ≥ 0. This bell-like shape rises from zero at x = 0, peaks at x = σ, and then falls away asymptotically toward zero. The cumulative distribution function (CDF) is F(x; σ) = 1 − exp(−x²/(2σ²)), which gives the probability that a random observation falls at or below x. The complementary CDF (CCDF = 1 − CDF) gives the probability of observing a value strictly greater than x — this is often called the survival function and is critical in reliability and communications engineering.
The Rayleigh distribution is a special case of the two-parameter Weibull distribution with shape parameter k = 2. It also has a deep connection to the normal distribution: if two independent random variables X and Y each follow a zero-mean normal distribution with variance σ², then the vector magnitude R = √(X² + Y²) follows a Rayleigh distribution with scale parameter σ. This geometric interpretation makes it the natural model for the amplitude of a 2D random vector.
In wireless communications, the Rayleigh fading model describes how radio signals propagate in environments with many scatterers and no dominant line-of-sight path. When a transmitted signal reflects off buildings, vehicles, and terrain before reaching the receiver, the envelope of the received signal follows a Rayleigh distribution. Engineers use this model to compute link budgets, determine outage probabilities, and design error-correcting codes. The parameter σ is estimated from signal measurements and feeds directly into system-level simulations.
In oceanography and meteorology, the distribution models significant wave heights and peak wind speeds at a site. By fitting σ to historical data, engineers and scientists can estimate the probability of extreme events — for example, the probability that wave height exceeds a safety threshold during a 50-year storm. Similar applications appear in offshore platform design, coastal flood modelling, and wind turbine siting.
In reliability engineering, the Rayleigh distribution serves as a life-time distribution for components subject to cumulative damage from multiple independent stress factors. Unlike the exponential distribution, the Rayleigh hazard rate increases linearly with time (h(t) = t/σ²), meaning older components fail at higher rates — a realistic model for wear-out mechanisms such as metal fatigue and corrosion.
The key summary statistics are: Mean = σ√(π/2) ≈ 1.2533σ; Median = σ√(2 ln 2) ≈ 1.1774σ; Mode = σ; Variance = (4 − π)/2 · σ² ≈ 0.4292σ². The mean always exceeds the mode, reflecting the right skew of the distribution. The variance grows quadratically with σ, so doubling σ quadruples the spread.
Rayleigh distribution examples
Worked examples showing the PDF, CDF, and key statistics for different σ and x values.
| Inputs | Key Outputs | Application |
|---|---|---|
| σ = 1, x = 1 | PDF ≈ 0.6065, CDF ≈ 0.3935, Mean ≈ 1.2533 | Standard Rayleigh distribution. The mode equals σ = 1 and the mean is about 25% larger. |
| σ = 10, x = 12 | PDF ≈ 0.0584, CDF ≈ 0.5132, Mean ≈ 12.533 | Wind speed modeling. About 49% of observed wind speeds at this site will exceed 12 m/s. |
| σ = 5, x = 4 | PDF ≈ 0.1162, CDF ≈ 0.2739, Mean ≈ 6.267 | Signal envelope analysis. There is a 27.4% chance the signal amplitude is at or below 4 units. |
| σ = 1000, x = 800 | PDF ≈ 0.000581, CDF ≈ 0.2739, Mean ≈ 1253.3 | Reliability engineering. 72.6% of components survive beyond 800 hours with σ = 1000 h. |
How to use the Rayleigh Distribution Calculator
- Enter the scale parameter σ in the first field. σ must be a positive number; it equals the mode of the distribution and controls the overall spread.
- Enter the value x at which you want to evaluate the distribution. x must be zero or positive; negative values are outside the support of the distribution.
- Click Calculate. The tool instantly returns the PDF, CDF, complementary CDF, mean, median, mode, and variance.
- Read the CDF to find the probability that a random observation is ≤ x, or read the CCDF for the probability that it exceeds x.
- Click Reset to clear both fields, or load one of the example buttons to explore typical real-world parameter values.
Rayleigh Distribution FAQ
What is the scale parameter σ in the Rayleigh distribution?
σ is the sole parameter of the Rayleigh distribution. It equals the mode (the most probable value) of the distribution. A larger σ shifts the entire distribution to the right and increases its spread. In wireless communications σ is estimated from received-signal power measurements; in oceanography it is fitted to historical wave-height records.
How is the Rayleigh distribution related to the normal distribution?
If X and Y are independent zero-mean normal random variables each with variance σ², then the magnitude R = √(X² + Y²) follows a Rayleigh distribution with parameter σ. This is why the distribution arises naturally whenever you are interested in the 2D distance from the origin of a random point whose x- and y-coordinates are independent Gaussian noise.
What is the difference between the PDF and the CDF?
The PDF f(x) gives the probability density at a specific point — it describes how likely values near x are relative to other values. The CDF F(x) = P(X ≤ x) is the integral of the PDF from 0 to x and gives the probability that an observation falls at or below x. For the Rayleigh distribution, F(x) = 1 − exp(−x²/(2σ²)).
Why is the mean larger than the mode for the Rayleigh distribution?
The Rayleigh distribution is right-skewed: there is a long tail of high values that pulls the mean above the peak. The mean is σ√(π/2) ≈ 1.253σ while the mode is simply σ. The median σ√(2 ln 2) ≈ 1.177σ sits between them, as is typical for right-skewed distributions.
Can the Rayleigh distribution model wind speeds accurately?
The Rayleigh distribution is often used as a simplified model for wind speed in wind-energy assessments. It is a special case of the more general Weibull distribution with shape parameter k = 2. For sites where the wind-speed distribution is approximately symmetric around its peak, the Rayleigh model works well; otherwise, fitting a full Weibull distribution with its two parameters is preferred.
What is the complementary CDF (CCDF) and when do I use it?
The CCDF (or survival function) is 1 − F(x) = exp(−x²/(2σ²)) and gives the probability that an observation exceeds x. Engineers use it to compute outage probabilities (probability that signal strength drops below a threshold), exceedance probabilities in hydrology (probability that a flood level is exceeded), and survival fractions in reliability (fraction of components still working at time x).