The Rayleigh Distribution is a continuous probability distribution used to model the magnitude of a two-dimensional vector whose components are independent and normally distributed. It often appears in fields like signal processing, physics, and engineering.

With NumPy, you can easily simulate and analyze Rayleigh-distributed data.

What is the Rayleigh Distribution?

The Rayleigh distribution is a special case of the Weibull distribution and is commonly used to describe:

• Wind speed models

• Random signal amplitudes (e.g., radio signals)

• Scattering noise in radar systems

Probability Density Function (PDF)

f(x;σ)=xσ2e−x22σ2,x≥0f(x; \sigma) = \frac{x}{\sigma^2} e^{-\frac{x^2}{2\sigma^2}}, \quad x \geq 0

Where:

• xx: Random variable (≥ 0)

• σ\sigma: Scale parameter (controls spread)

Key Statistics

NumPy’s rayleigh() Function

numpy.random.Generator.rayleigh(scale=1.0, size=None)

Parameters

✅ Returns

An array of random values from a Rayleigh distribution.

✅ Example: Generate Rayleigh Data

import numpy as np

rng = np.random.default_rng(seed=42)

# Generate 1000 Rayleigh-distributed samples with scale σ = 2
data = rng.rayleigh(scale=2.0, size=1000)

print(data[:5])  # First 5 values

Visualizing the Rayleigh Distribution

import matplotlib.pyplot as plt
import seaborn as sns

sns.histplot(data, bins=30, kde=True, color='teal')
plt.title("Rayleigh Distribution (scale=2.0)")
plt.xlabel("Value")
plt.ylabel("Frequency")
plt.grid(True)
plt.show()

You’ll see a skewed distribution that peaks and tapers off.

Effect of Scale (σ)

Try generating data with different scale parameters:

scales = [0.5, 1.0, 2.0, 4.0]

for scale in scales:
    data = rng.rayleigh(scale=scale, size=1000)
    sns.kdeplot(data, label=f'scale={scale}')

plt.title("Rayleigh Distributions for Different Scales")
plt.xlabel("Value")
plt.ylabel("Density")
plt.legend()
plt.grid(True)
plt.show()

Observation:

• A larger σ stretches the curve wider (more spread).

• A smaller σ compresses it, making the peak sharper.

Real-Life Applications

Full Simulation Example: Signal Strengths

scale = 3.0  # Assume average signal strength follows Rayleigh(σ=3)
samples = rng.rayleigh(scale=scale, size=10000)

# Calculate mean and variance
print("Empirical Mean:", np.mean(samples))
print("Theoretical Mean:", scale * np.sqrt(np.pi / 2))

print("Empirical Variance:", np.var(samples))
print("Theoretical Variance:", (2 - np.pi/2) * scale**2)

# Plot histogram
sns.histplot(samples, bins=50, kde=True, color='orchid')
plt.title("Rayleigh Signal Strength Simulation (scale=3.0)")
plt.xlabel("Signal Strength")
plt.ylabel("Density")
plt.grid(True)
plt.show()

Tips for Using Rayleigh Distribution

⚠️ Common Pitfalls

Relationship with Other Distributions

Conclusion

The Rayleigh Distribution is a powerful and practical model for signal strengths, wave magnitudes, and radar systems. With NumPy, generating and analyzing Rayleigh data is simple, efficient, and effective for simulations and applied statistics.

Summary Table