1. What is the Rayleigh Distribution?

The Rayleigh distribution is a continuous probability distribution for non-negative valued random variables. It is often used when the magnitude of a vector is related to its directional components (e.g., wind speed, wave heights).

2. How Does the Calculator Work?

The calculator uses the Rayleigh probability density function:

Where:

• \( x \) — Value at which to evaluate the function (≥0)
• \( \sigma \) — Scale parameter of the distribution (>0)
• \( e \) — Euler's number (~2.71828)

Explanation: The function gives the relative likelihood of the random variable taking on the given value x.

3. Applications of Rayleigh Distribution

Details: The Rayleigh distribution is used in various fields including signal processing, oceanography, and physics. It describes the distribution of wind speeds, wave heights, and the magnitude of complex random variables.

4. Using the Calculator

Tips: Enter the value x (must be ≥0) and the scale parameter σ (must be >0). The calculator will compute the probability density at point x.

5. Frequently Asked Questions (FAQ)

Q1: What does the scale parameter σ represent?
A: The scale parameter σ determines the spread of the distribution. The mode occurs at x = σ, and the mean is σ√(π/2).

Q2: How is Rayleigh different from Normal distribution?
A: Rayleigh describes the magnitude of a 2D vector with normally distributed components, while Normal describes scalar quantities.

Q3: What is the relationship to chi-squared distribution?
A: A Rayleigh-distributed variable is the square root of an exponential random variable (which is chi-squared with 2 degrees of freedom).

Q4: When is Rayleigh distribution appropriate?
A: When dealing with magnitude measurements where orthogonal components are independent and normally distributed with equal variance.

Q5: What are typical values for σ?
A: In practical applications, σ depends on the context (e.g., for wind speeds, σ might range from 1-10 m/s depending on location).