1. What is the Weibull Distribution?

The Weibull distribution is a continuous probability distribution widely 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 cumulative distribution function (CDF):

Where:

• \( x \) — Value at which to evaluate the function (x ≥ 0)
• \( \lambda \) — Scale parameter (λ > 0)
• \( k \) — Shape parameter (k > 0)

Explanation: The CDF gives the probability that a Weibull-distributed random variable is less than or equal to x.

3. Importance of Weibull Distribution

Details: The Weibull distribution is extremely flexible and can model various types of failure rates (increasing, decreasing, or constant). It's used in fields like reliability engineering, weather forecasting, and medical survival analysis.

4. Using the Calculator

Tips: Enter the value x where you want to evaluate the CDF, the scale parameter λ, and the shape parameter k. All parameters must be positive 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: What are typical uses of Weibull distribution?
A: Reliability analysis, failure time modeling, wind speed distributions, and modeling extreme values.

Q3: How is the scale parameter (λ) interpreted?
A: λ is the 63.2nd percentile of the distribution (when x = λ, F(x) ≈ 0.632).

Q4: Can the Weibull distribution model bathtub curves?
A: The standard 2-parameter Weibull cannot, but modified versions (3-parameter Weibull) can model bathtub-shaped failure rates.

Q5: What's the relationship between Weibull and exponential distribution?
A: The exponential distribution is a special case of the Weibull distribution when k = 1.