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Chapter 4: Problem 4

The life of automobile voltage regulators has an exponential distribution with a mean life of 6 years. You purchase a 6-year-old automobile with a working voltage regulator and plan to own it for 6 years a. What is the probability that the voltage regulator fails during your ownership? b. If your regulator fails after you own the automobile 3 years and it is replaced, what is the mean time until the next failure?

Short Answer

Expert verified
a. Probability of failure is approximately 0.632. b. Mean time to next failure is 6 years.

Step by step solution

01

Understanding the Exponential Distribution

The exponential distribution is often used to model the time until an event occurs, like the failure of a component. It is defined by the probability density function (PDF): \( f(t) = \lambda e^{-\lambda t} \), where \( \lambda \) is the rate parameter. For an exponential distribution, \( \lambda = \frac{1}{\text{mean}} \). Since the mean life is 6 years, \( \lambda = \frac{1}{6} \).
02

Calculating Probability of Failure within Ownership Period (First Part)

To find the probability of failure during the ownership period of 6 years, we need to compute the probability that the voltage regulator fails within that time. This is given by the cumulative distribution function (CDF) for the exponential distribution: \( P(T \leq x) = 1 - e^{-\lambda x} \). Here, \( x = 6 \) and \( \lambda = \frac{1}{6} \). Hence, the probability is: \[ P(T \leq 6) = 1 - e^{-\frac{1}{6} \times 6} = 1 - e^{-1} \approx 0.632 \]
03

Understanding Memoryless Property (Second Part)

The exponential distribution has a memoryless property, meaning the probability distribution of the time until next failure does not depend on how much time has already elapsed. If the regulator fails after 3 years and is replaced, the mean time to the next failure is the same as the original mean, which is 6 years.

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Key Concepts

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

Probability Density Function
The probability density function (PDF) is a fundamental concept in probability and statistics, used to describe the likelihood of a continuous random variable. For an exponential distribution, the PDF is represented by the equation:\[ f(t) = \lambda e^{-\lambda t} \]Here, \( \lambda \) is the rate parameter, and \( t \) represents time.
  • The value of \( \lambda \) is the reciprocal of the mean, so with a mean life of 6 years, \( \lambda = \frac{1}{6} \).
  • The PDF shows how probabilities are distributed over different possible values of \( t \).
  • It indicates the instantaneous rate of failure per unit time at each point \( t \).
The exponential PDF is useful in modeling scenarios like the life span of electronic components, where events happen continuously and independently.
Cumulative Distribution Function
The cumulative distribution function (CDF) provides the probability that a random variable takes a value less than or equal to a specific point. For the exponential distribution, it is defined as:\[ P(T \le x) = 1 - e^{-\lambda x} \]Where \( x \) is the time by which the event might have occurred.
  • The CDF accumulates the probability from time zero to \( x \).
  • In our exercise, it forecasts the probability that a voltage regulator fails within 6 years.
  • This is calculated as \[ P(T \le 6) = 1 - e^{-\frac{1}{6} \times 6} = 1 - e^{-1} \approx 0.632 \], which means about a 63.2% chance of failure during ownership.
The CDF is a powerful tool for understanding the distribution of probability over a given time period.
Memoryless Property
A unique feature of the exponential distribution is its memoryless property. This means that the probability of an event occurring in the future is independent of any past events.
  • For example, if a voltage regulator lasts 3 years, the probability of its failure in the next 6 years remains unaffected by the previous duration.
  • This property implies that the exponential distribution does not "remember" past data.
In practical terms, after replacing a failed component, the expected life span of the new component remains the same as the original, say 6 years in this scenario. The memoryless property is crucial for reliability analysis and helps in simplifying complex calculations.
Mean Life
Mean life refers to the expected value or average time before a failure or event occurs. In the context of the exponential distribution, the mean life is particularly significant because it equals the reciprocal of the rate parameter \( \lambda \).
  • If \( \lambda = \frac{1}{6} \), it informs us that the mean life is 6 years.
  • Mean life provides a general sense of how long similar voltage regulators might last.
  • It is a crucial measure used by engineers and statisticians to predict the lifespan of components or systems.
Calculating and understanding the mean life aids in planning, such as when to anticipate replacement and ensuring the reliability of the system over time.

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