Weibull Distribution Calculator

What is the Weibull Distribution?

The Weibull distribution is a continuous probability distribution named after Swedish mathematician Waloddi Weibull. It's widely used in reliability engineering and failure analysis due to its flexibility in modeling various types of data.

Formulas and Their Meanings

1. Probability Density Function (PDF): \[f(x) = \frac{\beta}{\alpha} (\frac{x}{\alpha})^{\beta-1} e^{-(\frac{x}{\alpha})^\beta}\] Where \(\alpha\) is the scale parameter and \(\beta\) is the shape parameter.

2. Cumulative Distribution Function (CDF): \[F(x) = 1 - e^{-(\frac{x}{\alpha})^\beta}\] This gives the probability that a value is less than or equal to x.

3. Mean: \[E(X) = \alpha \Gamma(1 + \frac{1}{\beta})\] Where \(\Gamma\) is the gamma function.

4. Variance: \[Var(X) = \alpha^2 [\Gamma(1 + \frac{2}{\beta}) - (\Gamma(1 + \frac{1}{\beta}))^2]\]

5. Mode (for \(\beta > 1\)): \[Mode = \alpha (\frac{\beta - 1}{\beta})^{\frac{1}{\beta}}\]

6. Median: \[Median = \alpha (\ln 2)^{\frac{1}{\beta}}\]

Calculation Steps

1. Input the scale parameter (\(\alpha\)) and shape parameter (\(\beta\)).
2. Specify the range [X1, X2] for probability calculation.
3. Calculate the CDF at X1 and X2.
4. Subtract CDF(X1) from CDF(X2) to get P(X1 < X < X2).
5. Use the formulas above to calculate mean, mode, median, variance, and standard deviation.

Example Calculation

Let's calculate for \(\alpha = 2\), \(\beta = 3\), X1 = 1, and X2 = 3

1. P(1 < X < 3) = F(3) - F(1) \[= (1 - e^{-(\frac{3}{2})^3}) - (1 - e^{-(\frac{1}{2})^3}) \approx 0.7769\]
2. Mean: \[E(X) = 2 \cdot \Gamma(1 + \frac{1}{3}) \approx 1.7724\]
3. Mode: \[2 \cdot (\frac{3 - 1}{3})^{\frac{1}{3}} \approx 1.6984\]
4. Median: \[2 \cdot (\ln 2)^{\frac{1}{3}} \approx 1.7673\]
5. Variance: \[4 \cdot [\Gamma(1 + \frac{2}{3}) - (\Gamma(1 + \frac{1}{3}))^2] \approx 0.3679\]
6. Standard Deviation: \[\sqrt{0.3679} \approx 0.6066\]

Visual Representation

This graph represents a typical Weibull distribution. The shape can vary significantly based on the α and β parameters.