Rayleigh Distribution Calculator
Free Rayleigh distribution calculator. Compute PDF, CDF, P(X≤x), P(a≤X≤b), mean, median, mode, varia
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Rayleigh Distribution — Magnitude of 2D Gaussian Vectors
Wind speed, wave height, wireless fading. From 2D Gaussian components to real-world magnitude modeling. PDF, CDF, percentiles, P(a≤X≤b).
Real-World Scenarios — Click to Load
Input Parameters
PDF with P(a≤X≤b) Shaded
Moments Comparison
Percentiles
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- The Rayleigh distribution models the magnitude of a 2D vector whose components are independent N(0,σ²)
- PDF: f(x) = (x/σ²) × exp(-x²/(2σ²)) for x ≥ 0; CDF: F(x) = 1 - exp(-x²/(2σ²))
- Mean = σ√(π/2) ≈ 1.2533σ, Median = σ√(2ln2) ≈ 1.1774σ, Mode = σ
- Variance = σ²(4-π)/2 ≈ 0.4292σ²
- P(a ≤ X ≤ b) = exp(-a²/(2σ²)) - exp(-b²/(2σ²)); Percentile q: x_q = σ√(-2ln(1-q))
Did You Know?
How It Works
1. 2D Gaussian Magnitude
If Z₁, Z₂ ~ N(0,σ²) independent, then X = √(Z₁² + Z₂²) ~ Rayleigh(σ).
2. PDF and CDF
PDF f(x)=(x/σ²)exp(-x²/(2σ²)) peaks at x=σ. CDF F(x)=1-exp(-x²/(2σ²)) has closed form.
3. Mean, Median, Mode
Mean > Median > Mode (right-skewed). Mode = σ is the peak. Mean = σ√(π/2) ≈ 1.253σ.
4. Wireless Applications
Rayleigh fading: when signal arrives via many paths with random phase, no dominant component.
5. Percentiles
The q-th percentile is x_q = σ√(-2ln(1-q)). Median = σ√(2ln2).
Expert Tips
Rayleigh vs Rice
When there is a dominant (line-of-sight) component, use Rice distribution. Rayleigh = Rice with zero dominant.
σ Interpretation
σ is the scale. Larger σ = more spread, higher mean. Mode = σ always.
Weibull Connection
Rayleigh(σ) = Weibull(shape=2, scale=σ√2). Weibull generalizes Rayleigh.
Chi-Square Link
If X ~ Rayleigh(σ), then X²/(2σ²) ~ Exp(1). Squared Rayleigh relates to exponential.
Why Use This Calculator vs Other Tools?
| Feature | This Calculator | Excel | R/Python |
|---|---|---|---|
| PDF + CDF + Percentiles | ✅ | ❌ No built-in | ⚠️ Package needed |
| P(a≤X≤b) shaded region | ✅ | ❌ | ⚠️ Manual |
| 7 real-world examples | ✅ | ❌ | ❌ |
| Interactive charts | ✅ | ❌ | ⚠️ Requires plot |
| Copy & share results | ✅ | ❌ | ❌ |
| AI analysis | ✅ | ❌ | ❌ |
Frequently Asked Questions
When should I use the Rayleigh distribution?
When modeling the magnitude of a 2D vector whose components are independent N(0,σ²): wind speed magnitude, signal envelope in fading, radial error in 2D, wave height.
What is the relationship to the normal distribution?
If Z₁, Z₂ ~ N(0,σ²) independent, then X = √(Z₁²+Z₂²) ~ Rayleigh(σ). Rayleigh is the distribution of the magnitude.
What is Rayleigh fading?
In wireless, when signal arrives via many scattered paths with random phase (no line-of-sight), the envelope follows Rayleigh. Classic model for fading channels.
How do I interpret σ?
σ is the scale parameter. Mode = σ. Mean = σ√(π/2) ≈ 1.253σ. Larger σ means more spread and higher values.
Rayleigh vs Rice distribution?
Rice has a dominant (line-of-sight) component; Rayleigh has none. Rayleigh = Rice with zero dominant component.
How do I fit Rayleigh to data?
MLE: σ̂ = √(Σxᵢ²/(2n)). Or use method of moments: σ̂ = x̄/√(π/2).
Rayleigh by the Numbers
Official Data Sources
Disclaimer: This calculator uses exact closed-form formulas for the Rayleigh distribution. Results are mathematically exact. For critical applications (wireless system design, wind energy, structural engineering), verify assumptions (2D Gaussian components, independence). This tool is for educational and professional reference purposes.
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