Statistics & Probability

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Rayleigh Distribution Calculator.

Calculate Rayleigh PDF, CDF, mean, and variance from a scale parameter.

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01

Set your values

Results update as you type.

probabilityDensity: 0.283065

probabilityDensity

0.283065
cumulativeProbability: 0.24516

cumulativeProbability

0.245160
mean: 2.506628

mean

2.506628
variance: 1.716815

variance

1.716815

This route uses the standard scale parameter σ and x ≥ 0. It reports PDF and CDF at the entered x plus the distribution mean and variance; alternate parameterizations and quantiles are not inferred.

Results update automatically as you type.

Use Cases

Wind energy analysis

Engineers use the Rayleigh distribution to model wind speed variations at a site, helping estimate energy output and design wind turbines.

Example: Given σ=5 m/s, find the probability that wind speed is below 10 m/s.

Signal amplitude modeling

In wireless communications, the Rayleigh distribution describes the envelope of a received signal in multipath fading, aiding in link budget analysis.

Example: Calculate the mean signal amplitude for σ=2 volts.

Frequently Asked Questions

What is the Rayleigh distribution used for?
The Rayleigh distribution models the magnitude of a vector with two independent normal components of equal variance. It's commonly used in signal processing, wind speed analysis, and reliability engineering to describe the distribution of magnitudes or amplitudes.
How do I interpret the scale parameter σ?
The scale parameter σ determines the spread of the distribution. The mean is σ√(π/2) and the variance is (2 - π/2)σ². A larger σ shifts the distribution to higher values and increases variability.
What does the cumulative probability represent?
The cumulative probability (CDF) gives the probability that a random variable following the Rayleigh distribution is less than or equal to the specified value x. It is calculated as 1 - exp(-x²/(2σ²)).

Tips & Common Mistakes

Tips

  • Ensure σ is positive; the Rayleigh distribution is undefined for σ ≤ 0.
  • The value x must be non-negative; probabilities for negative x are zero.
  • Use consistent units for σ and x (e.g., both in meters or both in volts) to get meaningful results.
  • For quick estimates, remember the mean is about 1.253σ and the standard deviation is about 0.655σ.

Common Mistakes to Avoid

  • Entering a negative value for σ, which is invalid and will produce errors.
  • Confusing the scale parameter σ with the standard deviation; the standard deviation is σ√(2 - π/2), not σ.
  • Using x as the scale parameter instead of σ, leading to incorrect probability calculations.

Last updated: August 13, 2026