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Chapter 4: Problem 167

The CPU of a personal computer has a lifetime that is exponentially distributed with a mean lifetime of six years. You have owned this CPU for three years. (a) What is the probability that the CPU fails in the next three years? (b) Assume that your corporation has owned 10 CPUs for three years, and assume that the CPUs fail independently. What is the probability that at least one fails within the next three years?

Short Answer

Expert verified
(a) \(0.393\); (b) \(0.9968\).

Step by step solution

01

Understanding the Exponential Distribution

An exponential distribution is characterized by the rate parameter \( \lambda \), which is the reciprocal of the mean. In this exercise, the mean lifetime of the CPU is six years, so \( \lambda = \frac{1}{6} \) years.
02

Calculate the Probability of CPU Failing in Next Three Years

Since the CPU is already three years old, we want to find the probability that it will fail in the next three years given it has survived up to now. For an exponential distribution, this probability does not depend on how much lifetime it has already completed (memoryless property). The probability that the CPU fails in the next three years is given as \( 1 - e^{-\lambda \cdot 3} = 1 - e^{-\frac{1}{6} \times 3} = 1 - e^{-0.5} \approx 0.393\).
03

Setting Up Probability for 10 Independent CPUs

Now we have a scenario with 10 CPUs. We will use the complementary probability of all CPUs not failing within the next three years. This is calculated as \( (0.607)^{10} \), since each CPU has a probability of \( 0.607 \) not failing in the three-year period.
04

Calculate Probability at Least One CPU Fails

To find the probability that at least one CPU fails, we subtract the probability that none of the CPUs fail from 1. Thus, \( 1 - (0.607)^{10} \approx 1 - 0.0032 \approx 0.9968\).

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Key Concepts

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

Memoryless Property
The memoryless property is a unique feature of the exponential distribution. This property states that the probability of an event occurring in the future is independent of any past events. In simpler terms, the remaining lifetime of an object is unaffected by how long it has already been operating.

For this exercise, even though the CPU is three years old, the probability of it failing in the next three years remains unaffected, similar to a brand-new CPU. This is due to the memoryless property. It simplifies probability calculations as it eliminates the need to consider past durations.
Probability Calculation
Probability calculations involving exponential distributions often rely on the rate parameter. This helps in determining the likelihood of events within a specific time frame.

For exponentially distributed variables, we use the formula: \(P(T > t) = e^{- rac{t}{ ext{mean}}}\)
To find the probability that the CPU, with a mean life of six years, will fail within the next three years, we compute: \(1 - e^{- rac{1}{6} imes 3}\). After a few calculations, we realize that we have a 39.3% chance of failure within this period.
  • The initial part of the formula, \( e^{- rac{t}{6}} \), represents the likelihood of a CPU not failing by a certain time \( t \).
  • Subtracting from 1 gives us the probability that failure happens within the specified time.
Independent Events
When dealing with multiple objects in exponential distribution problems, such as multiple CPUs, independence can simplify calculations. Events are independent if the occurrence of one event does not affect the probability of another event occurring.

The example with 10 CPUs assumes that each CPU operates independently. This means the failure of one CPU doesn't influence the failure of others. To calculate probabilities with independent events, you often use complementary probability.
  • Calculate the probability of one event happening or not.
  • Raise this probability to the power of the number of independent events (CPUs, in this case).
  • Finally, adjust this probability to find the scenario of interest.
In our situation, the probability that none out of 10 CPUs fail is raised to the power of 10. The probability of at least one failing is found by subtracting this result from 1.
Rate Parameter
In exponential distributions, the rate parameter \( \lambda \) is crucial for calculations. It's essentially the inverse of the mean and dictates how quickly or slowly events (failures, in this case) occur.

For the CPU exercise, with a mean lifetime of six years, the rate parameter is: \( \lambda = \frac{1}{6} \).
Knowing \( \lambda \) allows us to compute probabilities for various time frames using the exponential probability formula. This parameter is vital because:
  • It directly influences the exponential decay function, dictating how rapidly the probability decreases over time.
  • It simplifies the process of understanding the expected time to an event, especially when comparing different scenarios with varying mean times.
By understanding \( \lambda \), we gain a clearer picture of how likely or unlikely events are to happen within certain intervals.

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