1. What is the Weibull Distribution?

The Weibull distribution is a continuous probability distribution used in reliability engineering, failure analysis, and survival analysis. It's particularly useful for modeling time-to-failure data.

2. How Does the Calculator Work?

The calculator uses the Weibull probability density function:

Where:

• \( k \) — Shape parameter (dimensionless)
• \( \lambda \) — Scale parameter (value)
• \( x \) — Value at which to evaluate the function (value)

Explanation: The shape parameter determines the failure rate behavior (increasing, constant, or decreasing). The scale parameter determines the spread of the distribution.

3. Applications of Weibull Distribution

Details: The Weibull distribution is widely used in reliability engineering to model failure times, in weather forecasting to describe wind speed distributions, and in industrial engineering to represent manufacturing and delivery times.

4. Using the Calculator

Tips: Enter positive values for shape (k) and scale (λ) parameters. The x value must be non-negative. All parameters must be valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: What does the shape parameter (k) indicate?
A: k < 1 indicates decreasing failure rate, k = 1 constant rate (exponential distribution), k > 1 increasing failure rate.

Q2: How is the scale parameter (λ) interpreted?
A: λ represents the characteristic life where approximately 63.2% of items will have failed.

Q3: What's the difference between PDF and CDF?
A: PDF gives the probability density at a point, while CDF gives the probability up to that point.

Q4: Can this calculator compute the cumulative distribution?
A: This version calculates the probability density function (PDF) only.

Q5: What industries use Weibull analysis most?
A: Aerospace, automotive, electronics, and medical device industries rely heavily on Weibull analysis.