HesapHanem

Rayleigh Distribution Calculator

Enter the scale parameter σ and a value x to evaluate the Rayleigh distribution.

How it is calculated

Enter the scale parameter σ (greater than zero) and a point x (zero or more). The tool returns the density at that point, f(x) = (x ÷ σ²) × e−x² ÷ (2σ²), and the cumulative probability, F(x) = 1 − e−x² ÷ (2σ²), which is the chance that a Rayleigh variable is at most x. It also gives the distribution's mean σ × √(π ÷ 2), variance (4 − π) ÷ 2 × σ² and mode σ, which depend only on σ. Density and cumulative probability are shown to six decimals, the other three to four. Note that σ is not the standard deviation of the Rayleigh variable itself; it is the scale of the two underlying normal components. A density value is not a probability and can exceed 1 for a small σ.

Worked example

Take σ = 2 and x = 3. First, x² ÷ (2σ²) = 9 ÷ 8 = 1.125, and e−1.125 = 0.324652. Density: (3 ÷ 4) × 0.324652 = 0.243489. Cumulative probability: 1 − 0.324652 = 0.675348, so there is about a 67.5% chance the value is 3 or less. Mean: 2 × √(π ÷ 2) = 2.5066. Variance: (4 − π) ÷ 2 × 4 = 1.7168. Mode: 2.0000, the peak of the curve, equal to σ.

Try this example →

Common mistakes

  • Treating σ as the standard deviation of the Rayleigh variable. It is the scale parameter, and the actual standard deviation is the square root of the variance, about 0.655 × σ.
  • Reading the density as a probability. The value 0.243489 is a height of the curve; use the cumulative figure for the chance of staying below x.
  • Using the cumulative value for the chance of exceeding x. That chance is 1 − F(x), here 0.324652.
  • Entering a negative x. The Rayleigh distribution only exists for zero and above, so the tool gives no result.

About this calculator

The Rayleigh distribution describes the magnitude of a 2D vector whose components are independent, zero-mean, equally-scaled normal random variables. Its probability density function is f(x) = (x/σ²) × e^(−x²/2σ²) and its cumulative distribution function is F(x) = 1 − e^(−x²/2σ²), with mean = σ√(π/2), variance = (4−π)/2 × σ², and mode = σ.

Wireless communications engineers use it to model signal amplitude fading in environments with many scattered, reflected signal paths and no dominant line-of-sight (Rayleigh fading), meteorologists use it to model wind speed distributions, and reliability engineers use it for failure-time analysis when a failure rate increases linearly with time.

Was this helpful?

You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    Sigma is the mode of a Rayleigh curve, not its average or its spread. With sigma 2 the tool reports a mode of 2.0000, a mean of 2.5066 and a variance of 1.7168, so a measured mean must be divided by about 1.2533 before it estimates sigma. The shape suits the length of a two-dimensional vector with equal, centred errors, such as wind speed components. What data are you trying to fit with it?

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators