Rayleigh Distribution Calculator
Calculate PDF, CDF, quantiles, and statistical properties of the Rayleigh distribution
Rayleigh Distribution Calculator
Must be positive (σ > 0)
Non-negative value (x ≥ 0)
Results
PDF Formula: f(x) = (x/σ²) × exp(-x²/(2σ²))
Parameters: σ = 1, x = 1
Probability Ranges
Example: Wind Speed Modeling
Wind Speed Analysis
Scale Parameter σ: 5.0 m/s (typical for moderate wind conditions)
Calculate PDF at x = 6 m/s:
f(6) = (6/5²) × exp(-6²/(2×5²)) = 0.240 × exp(-0.36) ≈ 0.168
Interpretation: Peak probability density for wind speeds around 6 m/s
Statistical Properties
Mean wind speed: 5 × √(π/2) ≈ 6.27 m/s
Most likely speed (mode): 5.0 m/s
Median wind speed: 5 × √(2 × ln(2)) ≈ 5.89 m/s
Rayleigh Distribution Properties
Scale Parameter
σ > 0, determines the spread
Mode
Mode = σ (most likely value)
Support
x ≥ 0 (non-negative values)
Common Applications
Wind speed modeling and analysis
Ocean wave height distribution
Electrical component lifetime analysis
Signal processing and communications
Quality control and reliability testing
Target shooting accuracy analysis
Understanding the Rayleigh Distribution
What is the Rayleigh Distribution?
The Rayleigh distribution is a continuous probability distribution named after Lord Rayleigh (John William Strutt). It's particularly useful for modeling phenomena where the magnitude of a vector composed of two orthogonal components with equal variance is of interest.
Key Characteristics
- •Support: x ≥ 0 (non-negative values only)
- •Single parameter: scale parameter σ > 0
- •Right-skewed distribution with mode at σ
- •Special case of the Weibull distribution
Mathematical Formulas
Probability Density Function (PDF)
f(x) = (x/σ²) × exp(-x²/(2σ²))
for x ≥ 0, σ > 0
Cumulative Distribution Function (CDF)
F(x) = 1 - exp(-x²/(2σ²))
Quantile Function
Q(p) = σ × √(-2 × ln(1-p))
Statistical Measures
Central Tendency
- Mean: σ√(π/2) ≈ 1.253σ
- Median: σ√(2ln(2)) ≈ 1.177σ
- Mode: σ
Variability
- Variance: σ²(4-π)/2 ≈ 0.429σ²
- Std. Dev: σ√((4-π)/2) ≈ 0.655σ
- Skewness: ≈ 0.631
Relationship to Normal
If X and Y are independent N(0,σ) random variables, then √(X² + Y²) follows a Rayleigh(σ) distribution.