Problem 163 - Applied Statistics and Probability for Engineers

Question

Suppose that the lifetime of a component (in hours) is modeled with a Weibull distribution with \(\beta=2\) and \(\delta=4000\). Determine the following in parts (a) and (b): (a) \(P(X>3000)\) (b) \(P(X>6000 \mid X>3000)\) (c) Comment on the probabilities in the previous parts compared to the results for an exponential distribution.

Step-by-Step Solution

Answer

(a) 0.5703, (b) 0.1848. Weibull shows declining reliability, unlike exponential.

Step 1: Understanding the Weibull Distribution

The probability density function of a Weibull distribution is given by \( f(x; \beta, \delta) = \beta \frac{x^{\beta-1}}{\delta^\beta} e^{-(x/\delta)^\beta} \), where \(\beta\) is the shape parameter and \(\delta\) is the scale parameter. We are given \(\beta = 2\) and \(\delta = 4000\). For part (a) and part (b), we first need the cumulative distribution function (CDF) \( F(x) = 1 - e^{-(x/\delta)^\beta} \).

Step 2: Calculating P(X > 3000)

To find \(P(X > 3000)\), we calculate the complementary cumulative distribution function: \( P(X > 3000) = 1 - F(3000) = e^{-(3000/4000)^2} \). Compute the exponent: \( (3000/4000)^2 = (0.75)^2 = 0.5625 \). Thus, \( P(X > 3000) = e^{-0.5625} \).

Step 3: Evaluating the Exponential Function

Using the value from the previous step, \( e^{-0.5625} \approx 0.5703 \). Therefore, \( P(X > 3000) \approx 0.5703 \).

Step 4: Conditional Probability P(X > 6000 | X > 3000)

For this conditional probability, \( P(X > 6000 | X > 3000) = \frac{P(X > 6000)}{P(X > 3000)} \). We need \( P(X > 6000) = e^{-(6000/4000)^2} \). Calculate the exponent: \( (6000/4000)^2 = (1.5)^2 = 2.25 \). Thus, \( P(X > 6000) = e^{-2.25} \).

Step 5: Evaluating P(X > 6000) and the Conditional Probability

Calculate \( e^{-2.25} \approx 0.1054 \). Apply the formula for conditional probability: \( P(X > 6000 | X > 3000) = \frac{0.1054}{0.5703} \approx 0.1848 \).

Step 6: Interpreting the Results

The probabilities derived from the Weibull distribution, \( P(X > 3000) \approx 0.5703 \) and \( P(X > 6000 | X > 3000) \approx 0.1848 \), imply decreasing reliability over time compared to an exponential distribution, where the conditional probability \( P(X > x+k | X > x) \) remains constant, due to the memoryless property, not present in the Weibull distribution.

Key Concepts

Probability Density Function

The concept of the Probability Density Function (PDF) is essential to understanding continuous distributions like the Weibull distribution. The PDF gives the likelihood of a random variable taking on a specific value. Although, in the case of continuous distributions, it is more about the density of values over an interval.

For the Weibull distribution, the PDF is defined as:

• \( f(x; \beta, \delta) = \beta \frac{x^{\beta-1}}{\delta^\beta} e^{-(x/\delta)^\beta} \)

Here, \( \beta \) is the shape parameter, and \( \delta \) is the scale parameter. The parameters affect how the distribution curves. The Weibull PDF is particularly flexible:

• When \( \beta < 1 \), it models a decreasing failure rate over time.
• When \( \beta = 1 \), it simplifies to the exponential distribution, indicating a constant failure rate.
• When \( \beta > 1 \), it depicts an increasing failure rate.

The PDF helps us understand the likelihood of a component surviving a specific amount of time, modeling the reliability of components in fields like engineering so they can prepare for maintenance schedules efficiently.

Cumulative Distribution Function

The Cumulative Distribution Function (CDF) is crucial for understanding probabilities within a range. Unlike the PDF, which gives information about density, the CDF gives cumulative probability up to a certain value.

For the Weibull distribution, the CDF is:

• \( F(x) = 1 - e^{-(x/\delta)^\beta} \)

The CDF is used to find:

• The probability that a random variable is less than or equal to a given value.

By subtracting the CDF value from 1, as shown in the exercise for \( P(X > 3000) \), you can find the probability that the random variable exceeds a particular value:

• \( P(X > 3000) = 1 - F(3000) = e^{-0.5625} \approx 0.5703 \)

The CDF is a powerful tool that allows us to calculate probabilities for continuous variables, helping engineers to assess risk over time.

Exponential Distribution

The Exponential Distribution is a special case of the Weibull distribution with a constant rate of events over time. This property is known as the "memoryless" property.

Key characteristics when \( \beta = 1 \):

• The PDF becomes \( f(x; \delta) = \frac{1}{\delta} e^{-x/\delta} \).
• It implies events (like failures) happen continuously and independently at a constant average rate.

In engineering, this means:

• Past failures do not influence future rates of failure.
• The probability of survival for the next unit of time is constant.

This differs from the Weibull with \( \beta eq 1 \), where the failure rate changes over time. Hence, the exponential distribution is useful when we can assume constant hazards, such as radioactive decay or certain life testing applications.

Conditional Probability

Conditional Probability is crucial when dealing with dependent events. In the context of the Weibull distribution, it's about finding the probability of an event, given that another event has already occurred.

For example, to determine \( P(X > 6000 \mid X > 3000) \) in the exercise:

• This represents the probability the component lasts 6000 hours, given it has already exceeded 3000 hours.
• Calculate it using \( \frac{P(X > 6000)}{P(X > 3000)} \).

For the Weibull distribution, conditional probabilities change as more time passes, reflecting that older components may have different reliabilities.

In contrast, with an exponential distribution, the memoryless property implies that the probability of survival beyond any additional time is the same, regardless of how long it has lasted so far. This is a defining difference that impacts decisions in real-world applications, such as planning and reliability testing. The absence of this property in Weibull necessitates more tailored strategies for asset management and lifecycle expectations.