Weibull Distribution Calculator Python

Weibull Distribution PDF:

\[ f(x; k, \lambda) = \frac{k}{\lambda} \left( \frac{x}{\lambda} \right)^{k-1} e^{-(x/\lambda)^k} \]

Probability Density (PDF):

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1. What is the Weibull Distribution?

The Weibull distribution is a continuous probability distribution commonly 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 computes the probability density function (PDF) using the formula:

\[ f(x; k, \lambda) = \frac{k}{\lambda} \left( \frac{x}{\lambda} \right)^{k-1} e^{-(x/\lambda)^k} \]

Where:

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

Explanation: The shape parameter determines the failure rate behavior, while the scale parameter stretches or compresses the distribution.

3. Importance 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 the value (x) at which you want to evaluate the PDF, the shape parameter (k), and the scale parameter (λ). All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What does the shape parameter (k) indicate?
A: When k < 1, failure rate decreases over time; k = 1 means constant failure rate (exponential distribution); k > 1 means increasing failure rate.

Q2: How is this different from normal distribution?
A: Weibull can model skewed data and has a more flexible shape, making it better for reliability data where normal distribution often fails.

Q3: What are typical applications of Weibull distribution?
A: Reliability analysis, failure prediction, wind speed modeling, and survival analysis in medical research.

Q4: How do I interpret the PDF value?
A: The PDF gives the relative likelihood of the random variable taking that exact value (not a probability, which is zero for continuous distributions).

Q5: Can this calculator compute cumulative probability?
A: This calculator only computes the PDF. For cumulative probabilities, you would need to calculate the CDF (cumulative distribution function).