Rayleigh Distribution

🌊 Rayleigh Distribution in Python

The Rayleigh distribution is a continuous probability distribution often used in signal processing, communications, and reliability analysis. It describes the magnitude of a vector with two independent, normally distributed components.

1. Characteristics of Rayleigh Distribution

Continuous distribution
Parameter: scale = σ (standard deviation of underlying normal distributions)
Values ≥ 0
PDF (Probability Density Function):

$\sigma) = \frac{x}{\sigma^2} e^{-x^2/(2\sigma^2)}, \quad x \ge 0$

Applications:

Modeling wind speed
Magnitude of noise in communication signals
Reliability of mechanical components

2. Generate Rayleigh Data Using NumPy

Output (example):

All values ≥ 0
Distribution is right-skewed

3. Visualize Rayleigh Distribution

Right-skewed
Peak occurs near scale

4. Compare Different Scale Parameters

data_small = np.random.rayleigh(scale=1.0, size=1000)
data_large = np.random.rayleigh(scale=3.0, size=1000)

sns.histplot(data_small, bins=30, kde=True, color='red', label='scale=1', alpha=0.5)
sns.histplot(data_large, bins=30, kde=True, color='blue', label='scale=3', alpha=0.5)
plt.title("Rayleigh Distribution Comparison")
plt.legend()
plt.show()

data_small = np.random.rayleigh(scale=1.0, size=1000)

data_large = np.random.rayleigh(scale=3.0, size=1000)

sns.histplot(data_small, bins=30, kde=True, color='red', label='scale=1', alpha=0.5)

sns.histplot(data_large, bins=30, kde=True, color='blue', label='scale=3', alpha=0.5)

plt.title("Rayleigh Distribution Comparison")

plt.legend()

plt.show()

Smaller scale → peak closer to 0
Larger scale → distribution spreads out

5. Compute Mean and Standard Deviation

Mean ≈ scale * √(π/2)
Standard deviation ≈ scale * √((4-π)/2)

🧠 Summary Table

Function	Parameters	Description
`np.random.rayleigh()`	scale, size	Generates Rayleigh random numbers
`scale`	Standard deviation	Determines spread of distribution
`size`	Number of samples	Output array size

🎯 Practice Exercises

Generate 1000 Rayleigh random numbers with scale=1.5 and plot histogram with KDE.
Compare Rayleigh distributions for scale=1, scale=2, scale=4.
Compute the mean and standard deviation and compare with theoretical formulas.

Rayleigh Distribution