91Ó°ÊÓ

The lifetime of a mechanical assembly in a vibration test is exponentially distributed with a mean of 400 hours. (a) What is the probability that an assembly on test fails in less than 100 hours? (b) What is the probability that an assembly operates for more than 500 hours before failure? (c) If an assembly has been on test for 400 hours without a failure, what is the probability of a failure in the next 100 hours? (d) If 10 assemblies are tested, what is the probability that at least one fails in less than 100 hours? Assume that the assemblies fail independently. (e) If 10 assemblies are tested, what is the probability that all have failed by 800 hours? Assume that the assemblies fail independently.

Step by step solution

01

Understand the Exponential Distribution

The lifetime of the mechanical assembly follows an exponential distribution with a mean of 400 hours, which means \( \lambda = \frac{1}{400} \). The probability density function (PDF) of the exponential distribution is \( f(t) = \lambda e^{-\lambda t} \), and the cumulative distribution function (CDF) is \( F(t) = 1 - e^{-\lambda t} \).

02

Calculate Probability for Part (a)

To find the probability that an assembly fails in less than 100 hours, use the CDF: \( F(100) = 1 - e^{-\frac{100}{400}} = 1 - e^{-0.25} \). Computing this gives \( F(100) \approx 0.2212 \).

03

Calculate Probability for Part (b)

To find the probability of operating for more than 500 hours before failure, use the complement of the CDF: \( P(T > 500) = 1 - F(500) = e^{-\frac{500}{400}} = e^{-1.25} \). Computing this gives \( P(T > 500) \approx 0.2865 \).

04

Calculate Conditional Probability for Part (c)

Given an assembly has operated for 400 hours, the probability of failure in the next 100 hours is \( F(500) - F(400) \). So, using the CDF, calculate: \( F(500) - F(400) = (1 - e^{-1.25}) - (1 - e^{-1}) = e^{-1} - e^{-1.25} \). This is approximately \( 0.2325 \).

05

Calculate Probability for Part (d)

For 10 assemblies, where each has a probability of failing in less than 100 hours \( F(100) \approx 0.2212 \), use the binomial probability: \( P( \text{at least one fails} ) = 1 - (1 - 0.2212)^{10} \). This gives \( P \approx 0.9122 \).

06

Calculate Probability for Part (e)

For all 10 assemblies to fail by 800 hours, compute the complementary case of one assembly not failing: \( (F(800))^{10} \). Compute \( F(800) = 1 - e^{-2} \approx 0.8647 \), then \( 0.8647^{10} \approx 0.1699 \).

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Key Concepts

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

Probability Calculations

Probability calculations are essential in understanding how likely an event is to occur, especially in the context of exponential distribution. The exponential distribution is often used to model the time until an event occurs, such as the failure of a mechanical assembly in a vibration test. It only depends on the mean lifetime (rate parameter) of the event. This feature is useful in several real-world applications.

• To calculate the probability of a failure within a specific time frame, the cumulative distribution function (CDF) of the exponential distribution is used, represented as: \( F(t) = 1 - e^{-\lambda t} \).
• The mean lifetime, or the rate parameter \( \lambda \), is crucial for these calculations, derived as the reciprocal of the mean, i.e., \( \lambda = \frac{1}{\text{mean}} \).

With the exponential distribution, you can find probabilities for events occurring within or beyond a certain time range, leading to topics like failure analysis and reliability engineering.

Failure Analysis

Failure analysis involves understanding how and when a system might fail, relying heavily on probability calculations. In the context of the exponential distribution, a common approach is to evaluate the time until failure. If you know the mean lifetime of a component, you can determine the likelihood that it will fail within a specific duration. For instance, in vibration tests on mechanical assemblies, the exponential distribution helps engineers predict failures, important for planning maintenance schedules and improving designs.

• It allows engineers to identify potential weaknesses in assemblies and systems by calculating probabilities of failures within given intervals.
• Understanding how a system behaves under certain test conditions is key, which helps avoid unexpected downtimes and improve overall system design and performance.

Effective failure analysis with exponential distribution ensures that systems are robust and less prone to unexpected errors.

Conditional Probability

Conditional probability deals with finding the probability of an event occurring, given that another event has already occurred. In exponential distribution, you often calculate the probability of a failure given that an assembly has already survived for a certain period without failing. For example, if a mechanical assembly has lasted 400 hours without a failure in a vibration test, and you want to know the probability of it failing in the next 100 hours, this is where conditional probability comes in.

• Using conditional probability, you can update the likelihood of future outcomes given the past performance, which is crucial in decision-making processes.
• The calculation makes use of the existing cumulative distribution data to determine the probability for a specific time interval.

By considering prior conditions, engineers can better assess risk and make informed maintenance or replacement decisions.

Reliability Engineering

Reliability engineering focuses on the probability of a system or component to perform its required functions under stated conditions for a specified period. In the realm of exponential distribution, reliability engineering ensures systems meet performance standards without unexpected failures. Exponential distribution helps reliability engineers predict system behavior concerning its lifetime and failure rates. With tools like probability calculations, these engineers can ensure systems are designed and operated within safe working limits.

• Key activities include setting and improving production processes based on lifetime data and minimizing the risk of failure through design improvements.
• Reliability engineering also involves testing systems like assemblies under various conditions, observing how and when failures might occur.

By understanding these probabilities and conditions, reliability engineers can build systems that are not only efficient but also safe and sustainable over time.