1. What is the Weibull Distribution Mean?

The Weibull distribution is a continuous probability distribution commonly used in reliability engineering, failure analysis, and survival analysis. The mean of the Weibull distribution provides the expected value or average of the distribution.

2. How Does the Calculator Work?

The calculator uses the Weibull mean formula:

Where:

• \( \lambda \) — Scale parameter (value)
• \( k \) — Shape parameter (dimensionless)
• \( \Gamma \) — Gamma function (dimensionless)

Explanation: The formula calculates the expected value of the Weibull distribution by combining the scale parameter with the gamma function of the shape parameter.

3. Importance of Weibull Distribution

Details: The Weibull distribution is widely used to model reliability data and failure times. Understanding its mean helps in predicting average lifetimes of products or components.

4. Using the Calculator

Tips: Enter positive values for both scale and shape parameters. The shape parameter determines the distribution's behavior (exponential when k=1, Rayleigh when k=2).

5. Frequently Asked Questions (FAQ)

Q1: What does the scale parameter represent?
A: The scale parameter (λ) represents the characteristic life where approximately 63.2% of items have failed.

Q2: How does shape parameter affect the distribution?
A: k<1 indicates decreasing failure rate, k=1 constant rate (exponential), k>1 increasing failure rate.

Q3: What are typical applications of Weibull distribution?
A: Reliability analysis, wind speed distributions, failure analysis, and survival studies.

Q4: What's the relationship with exponential distribution?
A: When k=1, Weibull reduces to exponential distribution with mean=λ.

Q5: How accurate is the gamma function approximation?
A: The Lanczos approximation used here is accurate to about 15 decimal places.