What is the Rayleigh distribution formula?

The PDF is f(x) = (x/sigma^2) times exp(-x^2/(2 sigma^2)) for x at least 0. The CDF is F(x) = 1 minus exp(-x^2/(2 sigma^2)). The survival function is P(X greater than x) = exp(-x^2/(2 sigma^2)). Sigma is the scale parameter and also equals the mode.

What is the mean of the Rayleigh distribution?

The mean is mu = sigma times sqrt(pi/2), approximately 1.2533 times sigma. For sigma = 2, the mean is about 2.507. The variance is sigma^2 times (4 minus pi) divided by 2, approximately 0.4292 times sigma^2.

What is the median of the Rayleigh distribution?

The median is sigma times sqrt(ln 4), approximately 1.1774 times sigma. For sigma = 1 the median is about 1.177. It is always less than the mean (1.2533 sigma), consistent with the right skew.

What is the mode of the Rayleigh distribution?

The mode equals the scale parameter sigma exactly. It is the most likely value and the peak of the PDF. For sigma = 3 the peak of the distribution is at x = 3.

When should I use the Rayleigh distribution?

Use the Rayleigh distribution when a quantity is the magnitude of a 2-D vector whose two independent components are each normally distributed with mean zero and the same standard deviation sigma. Classic applications include wind speed, ocean wave heights, wireless channel fading, and radar target detection.

How is the Rayleigh distribution related to the normal distribution?

If X and Y are independent N(0, sigma^2) random variables, then the distance R = sqrt(X^2 + Y^2) follows a Rayleigh distribution with parameter sigma. The Rayleigh is the radial component of a 2-D Gaussian and naturally appears whenever you measure the magnitude of a 2-D noise vector.

What is the relationship between the Rayleigh and chi-squared distributions?

If X and Y are independent standard normals, then (X^2 + Y^2) follows a chi-squared distribution with 2 degrees of freedom. The square root of a chi-squared(2) random variable, scaled by sigma, gives a Rayleigh(sigma). Equivalently, (R/sigma)^2 is chi-squared(2) or Exponential(1/2).

What is Rayleigh fading in wireless communications?

Rayleigh fading models the envelope of a received radio signal that travels multiple reflected paths with no dominant line-of-sight component. Each multipath component contributes a Gaussian-distributed in-phase and quadrature signal, so the combined envelope is Rayleigh distributed. It predicts worst-case channel conditions and is used in LTE and 5G link budget analysis.

Rayleigh Distribution Calculator

Find Rayleigh distribution probabilities, PDF, CDF, mean, variance, and full distributional statistics for any scale parameter sigma.

📡 What is the Rayleigh Distribution?

The Rayleigh distribution is a continuous probability distribution that describes the magnitude of a two-dimensional vector whose components are each independent, zero-mean Gaussian random variables with equal variance. Named after Lord Rayleigh who used it in acoustics research in 1880, it arises naturally whenever you measure the distance from the origin to a point scattered by 2-D Gaussian noise.

The distribution has a single parameter, the scale parameter sigma, which equals the mode (most likely value). Real-world applications span wireless communications (Rayleigh fading channel models for LTE and 5G), meteorology (wind speed distributions where horizontal components are Gaussian), oceanography (wave height statistics), radar signal processing (target amplitude models), and reliability engineering (bearing failure times under combined stress). Whenever a physical quantity is the combined magnitude of two equal-variance, uncorrelated Gaussian forces, the Rayleigh distribution is the natural model.

A common misconception is that the Rayleigh distribution is simply an exponential distribution with a squared argument. While the CDF of a Rayleigh(sigma) and an Exponential(1/(2*sigma^2)) look similar when x^2 is substituted, they are different families. The Rayleigh PDF f(x) = (x/sigma^2) * exp(-x^2/(2*sigma^2)) has an extra factor of x that tilts the distribution away from zero. This produces a unimodal shape with a clear peak at x = sigma rather than the strictly decreasing shape of the exponential.

The Rayleigh distribution is a special case of the Weibull distribution (shape k = 2, scale lambda = sigma * sqrt(2)), and its square follows an exponential distribution with rate 1/(2*sigma^2). Understanding these relationships allows statisticians to use the well-developed Weibull and exponential toolkits for Rayleigh data. This calculator handles both the probability evaluation mode (PDF, CDF, survival for any x) and the full distributional statistics mode (mean, median, mode, IQR, skewness, kurtosis) so you can characterize any Rayleigh population instantly.

📐 Formula

f(x; σ) = (x / σ²) × e^{−x² / (2σ²)}

f(x; σ) = probability density function at value x

x = non-negative query value (x ≥ 0)

σ = scale parameter (sigma > 0); also the mode

CDF: P(X ≤ x) = 1 − e^{−x²/(2σ²)}

Survival: P(X > x) = e^{−x²/(2σ²)}

Mean: μ = σ × √(π/2) ≈ 1.2533 × σ

Median: σ × √(ln 4) ≈ 1.1774 × σ

Variance: σ² × (4 − π) / 2 ≈ 0.4292 × σ²

Skewness: 2√π(π − 3) / (4 − π)^3/2 ≈ 0.6311

Quantile: Q(p) = σ × √(−2 ln(1 − p))

Example: For σ = 2 and x = 2, CDF = 1 − e^−4/8 = 1 − e^−0.5 ≈ 0.3935 = 39.35%

📖 How to Use This Calculator

Steps

Choose a calculation mode - Select "Calculate Probability" to evaluate PDF, CDF, and survival for a specific x, or select "Distribution Stats" to get the complete parameter set for a given sigma.

Enter the scale parameter sigma - Type a positive value or drag the slider. Sigma controls the spread and equals the mode (peak) of the distribution. For wind speed in m/s, a typical sigma is 2 to 5.

Enter the query value x (Probability mode) - Set x to the threshold you want to evaluate. All x values of 0 or greater are valid. The calculator instantly returns the CDF, survival function, and PDF.

Read the distributional statistics - Mean, variance, standard deviation, median, and mode appear with every probability result. Switch to Distribution Stats mode for Q1, Q3, IQR, skewness, and excess kurtosis.

💡 Example Calculations

Example 1 - Wind Speed Modeling (sigma = 3 m/s, x = 4 m/s)

What is the probability that wind speed exceeds 4 m/s given sigma = 3 m/s?

Compute the exponent: x^2 / (2 * sigma^2) = 16 / (2 * 9) = 16/18 = 0.8889.

Survival: P(X > 4) = exp(-0.8889) = 0.4111, so about 41.11% of the time wind exceeds 4 m/s.

CDF: P(X ≤ 4) = 1 - 0.4111 = 0.5889. Mean wind = 3 * sqrt(pi/2) = 3 * 1.2533 = 3.76 m/s.

Survival P(X > 4 m/s) = 41.11%

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Example 2 - Wireless Fading (sigma = 1, signal threshold x = 1.5)

Rayleigh fading channel: probability that signal amplitude falls below threshold 1.5 with sigma = 1?

CDF = 1 - exp(-x^2 / (2 * sigma^2)) = 1 - exp(-2.25 / 2) = 1 - exp(-1.125) = 1 - 0.3247 = 0.6753.

PDF at x = 1.5: f(1.5) = (1.5/1) * exp(-1.125) = 1.5 * 0.3247 = 0.4870.

Interpreting: 67.53% of signal samples fall below the threshold 1.5 in a Rayleigh fading environment.

P(amplitude ≤ 1.5) = 67.53%

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Example 3 - Ocean Wave Height Statistics (sigma = 2 m)

Full distributional statistics for ocean wave heights with sigma = 2 m.

Mean = 2 * sqrt(pi/2) = 2 * 1.2533 = 2.507 m. Median = 2 * sqrt(ln 4) = 2 * 1.1774 = 2.355 m. Mode = sigma = 2.000 m.

Variance = 4 * (4 - pi) / 2 = 4 * 0.4292 = 1.717 m^2. SD = sqrt(1.717) = 1.310 m.

Q1 = 2 * sqrt(-2*ln(0.75)) = 2 * 0.7585 = 1.517 m. Q3 = 2 * sqrt(-2*ln(0.25)) = 2 * 1.6651 = 3.330 m. IQR = 1.813 m.

Mean wave height = 2.507 m, IQR = 1.813 m

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❓ Frequently Asked Questions

What is the Rayleigh distribution used for in engineering?+

The Rayleigh distribution models the amplitude of a signal or physical quantity that results from two independent, equal-variance Gaussian components. It is used in wireless channel modeling (Rayleigh fading), wind energy assessment (wind speed histograms), radar signal processing, ocean wave height statistics, and bearing lifetime analysis under combined radial and axial loads.

What is the scale parameter sigma and how do I estimate it?+

Sigma is the scale parameter that controls the spread of the distribution and also equals the mode (most likely value). Given a dataset, the maximum likelihood estimator is sigma = sqrt(sum of x_i^2 divided by 2n), where n is the sample size. Alternatively, sigma equals the mean of the sample divided by sqrt(pi/2), approximately 0.7979 times the sample mean.

How is the Rayleigh distribution related to the Weibull distribution?+

The Rayleigh distribution is a special case of the Weibull distribution with shape parameter k = 2 and scale parameter lambda = sigma times sqrt(2). Because the Weibull family includes both exponential (k=1) and Rayleigh (k=2) as special cases, Weibull regression tools can directly model Rayleigh data by fixing or estimating the shape parameter near 2.

What is the skewness of the Rayleigh distribution?+

The skewness is 2*sqrt(pi)*(pi - 3) divided by (4 - pi)^(3/2), approximately 0.6311 for any sigma. This positive skewness means the tail extends to the right and the mean exceeds the median, which exceeds the mode. The skewness is a fixed constant independent of the scale, making it a quick identifier of Rayleigh-distributed data.

How do I compute the Rayleigh CDF by hand?+

The CDF is P(X at most x) = 1 minus exp(-x^2 / (2*sigma^2)). Compute the ratio x^2 / (2*sigma^2), evaluate the negative exponential, then subtract from 1. For example, with sigma = 2 and x = 3: ratio = 9/8 = 1.125, exp(-1.125) = 0.3247, CDF = 1 - 0.3247 = 0.6753, or 67.53%.

What is the probability that a Rayleigh variable exceeds its mean?+

The mean is mu = sigma*sqrt(pi/2). Substituting into the survival function: P(X greater than mu) = exp(-mu^2 / (2*sigma^2)) = exp(-(sigma^2*pi/2) / (2*sigma^2)) = exp(-pi/4) approximately equals exp(-0.7854) approximately equals 0.4559, or about 45.6%. This is independent of sigma, so slightly less than half of all observations exceed the mean.

How does the Rayleigh distribution differ from the exponential distribution?+

Both are right-skewed continuous distributions on [0, infinity), but the exponential has a monotonically decreasing PDF that peaks at x = 0, while the Rayleigh PDF starts at 0, rises to a peak at x = sigma, then decreases. If Y is Rayleigh(sigma), then Y^2 is exponentially distributed with rate 1/(2*sigma^2). The exponential has the memoryless property; the Rayleigh does not.

What is the IQR of the Rayleigh distribution?+

The IQR is Q3 minus Q1 where Q(p) = sigma*sqrt(-2*ln(1-p)). For sigma = 1: Q1 = sqrt(-2*ln(0.75)) = 0.7585, Q3 = sqrt(-2*ln(0.25)) = 1.6651, IQR = 0.9066. For any sigma, IQR = 0.9066*sigma. This scales linearly with sigma.

Can the Rayleigh distribution have negative values?+

No. The Rayleigh distribution is defined only for x at least 0. Its PDF equals zero for all negative x. This makes it appropriate for modeling non-negative physical quantities like distances, speeds, signal amplitudes, and wave heights where negative values have no physical meaning.

How do I fit a Rayleigh distribution to real data?+

The maximum likelihood estimate (MLE) of sigma is: sigma_hat = sqrt(sum of x_i^2 divided by 2n). Compute the sum of squares of all n observations, divide by 2n, then take the square root. This estimator is unbiased and efficient. For wind speed data in m/s, typical fitted sigma values range from 2 to 6 depending on location and season.

What is the excess kurtosis of the Rayleigh distribution?+

The excess kurtosis is -(6*pi^2 - 24*pi + 16) / (4-pi)^2, approximately 0.2451. This is positive (leptokurtic), meaning the Rayleigh distribution has heavier tails than a normal distribution with the same variance. The value is a fixed constant independent of sigma, like the skewness.

What is the survival function of the Rayleigh distribution and why is it useful?+

The survival function (reliability function) is S(x) = P(X greater than x) = exp(-x^2 / (2*sigma^2)). It gives the probability that the variable exceeds threshold x. In reliability engineering, S(t) is the probability that a component survives past time t. In wireless communications, S(r) is the probability that the received signal amplitude exceeds a minimum detection threshold r.

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📌 Quick Tips

💡The scale parameter sigma is the mode of the Rayleigh distribution. Set sigma equal to the most common observed value in your dataset for a quick starting estimate.

💡The mean (sigma times sqrt(pi/2)) is always larger than the mode (sigma), reflecting the right-skewed shape. For wind speed data, the mean wind is about 25% higher than the modal wind.

💡Use the Distribution Stats mode to get Q1, Q3, and IQR at once. The IQR spans roughly 0.76 sigma to 1.67 sigma regardless of the scale, so the distribution is moderately spread.

💡The survival function P(X greater than x) equals exp(-x squared divided by 2 sigma squared). For wireless fading, this gives the probability that signal amplitude exceeds a threshold directly.

💡Rayleigh fading in wireless communications assumes the received signal envelope follows Rayleigh(sigma). The average received power is proportional to 2 sigma squared.