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The authors of the article "A Probabilistic Insulation Life Model for Combined Thermal-Electrical Stresses" (IEEE Trans. on Elect. Insulation, 1985: 519-522) state that "the Weibull distribution is widely used in statistical problems relating to aging of solid insulating materials subjected to aging and stress." They propose the use of the distribution as a model for time (in hours) to failure of solid insulating specimens subjected to \(\mathrm{AC}\) voltage. The values of the parameters depend on the voltage and temperature; suppose \(\alpha=2.5\) and \(\beta=200\) (values suggested by data in the article). a. What is the probability that a specimen's lifetime is at most 250? Less than 250? More than 300? b. What is the probability that a specimen's lifetime is between 100 and 250 ? c. What value is such that exactly \(50 \%\) of all specimens have lifetimes exceeding that value?

Step by step solution

01

Understanding the Weibull Distribution

The Weibull distribution is used to model life data and is characterized by two parameters: scale (\(\beta\)) and shape (\(\alpha\)). The probability density function is given by \( f(t; \alpha, \beta) = \frac{\alpha}{\beta} \left(\frac{t}{\beta}\right)^{\alpha - 1} e^{-\left(\frac{t}{\beta}\right)^{\alpha}} \) for \( t \geq 0 \). The cumulative distribution function (CDF) is \( F(t) = 1 - e^{-\left(\frac{t}{\beta}\right)^{\alpha}} \).

02

Calculating Probability at Most 250 Hours

To find \( P(T \leq 250) \), we calculate the CDF at \( t = 250 \). Using the given \( \alpha = 2.5 \) and \( \beta = 200 \), substitute into the CDF: \[ F(250) = 1 - e^{-\left(\frac{250}{200}\right)^{2.5}}. \] Evaluate the expression to find the probability.

03

Calculating Probability Less Than 250 Hours

Since the questions asks for the time strictly less than 250 hours, acknowledge that \( P(T < 250) = P(T \leq 250) \) because it's a continuous distribution, so the probability at a single point (250) is zero.

04

Calculating Probability More Than 300 Hours

We want \( P(T > 300) = 1 - P(T \leq 300) \). First, calculate \( P(T \leq 300) \) using the CDF: \[ F(300) = 1 - e^{-\left(\frac{300}{200}\right)^{2.5}}. \] Then subtract this result from 1.

05

Calculating Probability Between 100 and 250 Hours

To find \( P(100 < T < 250) \), calculate \( F(250) \) and \( F(100) \), then find the difference: \( P(100 < T < 250) = F(250) - F(100) \). Calculate \[ F(100) = 1 - e^{-\left(\frac{100}{200}\right)^{2.5}} \] and then subtract from \( F(250) \).

06

Finding the Median Lifetime

For the median lifetime where exactly 50% of specimens exceed a certain value \( M \), solve \( F(M) = 0.5 \). Set the CDF to 0.5: \[ 0.5 = 1 - e^{-\left(\frac{M}{200}\right)^{2.5}}. \] This simplifies to \( e^{-\left(\frac{M}{200}\right)^{2.5}} = 0.5 \). Solve for \( M \) by taking the log and isolating \( M \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Density Function

In statistics, the Probability Density Function (PDF) is a fundamental concept. It describes the likelihood of a random variable to take on a specific value. In the context of the Weibull distribution, the probability density function helps us understand the distribution of lifetimes of specimens subjected to stress.
For the Weibull distribution, the PDF is given by: \[ f(t; \alpha, \beta) = \frac{\alpha}{\beta} \left(\frac{t}{\beta}\right)^{\alpha - 1} e^{-\left(\frac{t}{\beta}\right)^{\alpha}} \]Here:

• \( \alpha \) is the shape parameter
• \( \beta \) is the scale parameter
• \( t \) is the time or the lifetime of the specimen

This equation shows how the probability varies with time. Understanding the PDF allows us to calculate the probability of a failure within any given time frame.

Cumulative Distribution Function

The Cumulative Distribution Function (CDF) builds on the PDF to give the probability that a random variable is less than or equal to a certain value. In simpler terms, it tells us the probability that a specimen will "survive" up to a certain point in time.
The CDF for the Weibull distribution is expressed as:\[ F(t) = 1 - e^{-\left(\frac{t}{\beta}\right)^{\alpha}} \]

• For any time \( t \), \( F(t) \) provides the probability that a specimen's lifetime does not exceed \( t \).
• For instance, \( F(250) \) would give you the probability that a specimen lasts at most 250 hours.

CDFs are crucial for determining probabilities over intervals, and they are used extensively in lifetime modeling.

Lifetime Modeling

Modeling the lifetime of materials, especially those exposed to stresses, is essential in fields like engineering and quality assurance. The Weibull distribution is particularly suited for this task, thanks to its flexibility and applicability to aging materials.
In lifetime modeling, the shape parameter \( \alpha \) indicates the failure rate:

• \( \alpha < 1 \): decreasing failure rate, indicating that the probability of failure decreases over time
• \( \alpha = 1 \): constant failure rate, similar to the exponential distribution
• \( \alpha > 1 \): increasing failure rate, suggesting that the probability of failure increases over time

The scale parameter \( \beta \) relates to the time scale over which failures occur. Larger values of \( \beta \) suggest a longer lifetime. By using these parameters, the Weibull distribution accurately models and analyzes the time to failure for insulating materials.

Statistical Problems

Statistical problems involving lifetime data often require us to make predictions about material durability. With the Weibull distribution, we can readily address questions about reliability and failure rates during the analysis of engineering materials.
Common statistical queries might include:

• Calculating the probability of failure before a certain time.
• Estimating how many specimens will fail within a specific time frame.
• Determining safe operating conditions based on expected lifetime.

By utilizing the Weibull distribution to tackle these questions, predictions are both robust and reliable, essential for both manufacturing and quality control.

Median Lifetime

The median lifetime is a statistic that indicates the time by which 50% of the specimens have failed. For Weibull-distributed lifetimes, finding the median involves solving the equation: \[ F(M) = 0.5 \]Here: \[ 0.5 = 1 - e^{-\left(\frac{M}{200}\right)^{2.5}} \]Solving this results in:

• Setting \( e^{-\left(\frac{M}{200}\right)^{2.5}} = 0.5 \)
• Taking the natural logarithm to simplify and solve for \( M \).

The median provides a valuable point of reference in assessing material reliability, helping predict how long insulation may last under given stress conditions.