The Weibull distribution is a statistical probability distribution widely used in various fields to model the time until a specific event occurs. It is named after the Swedish engineer Waloddi Weibull, who introduced it in the 1950s. The key characteristic of the Weibull distribution is its flexibility in capturing a wide range of shapes, making it applicable to a diverse range of data patterns. It is commonly used to analyze failure times or lifetimes of products and systems, such as the time until a machine breaks down or the lifespan of a particular component.
What distinguishes the Weibull distribution from other probability distributions is its ability to capture different types of failure patterns. It can represent a variety of shapes, including decreasing, constant, or increasing failure rates. This flexibility allows for a more accurate representation of real-world data, as failures in the real world do not always follow a specific pattern.
The Weibull distribution is particularly important because it provides valuable insights into the reliability and survival analysis of various systems. By estimating the parameters of the distribution from observed data, analysts can determine the failure rate over time, calculate the probability of failure within a given timeframe, and evaluate the reliability of a system or product.
Let's assume the following values:
Using the formula:
\[ P(X1 < X < X2) = e^{- \left(\frac{1}{2}\right)^{1.5}} - e^{- \left(\frac{3}{2}\right)^{1.5}} \] \[ = e^{-0.3536} - e^{-1.8371} \] \[ = 0.7022 - 0.1599 = 0.5423 \]
The probability P(X1 < X < X2) is 0.5423.
