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Chapter 5: Problem 93

Suppose two identical components are connected in parallel, so the system continues to function as long as at least one of the components does so. The two lifetimes are independent of each other, each having an exponential distribution with mean \(1000 \mathrm{~h}\). Let \(W\) denote system lifetime. Obtain the moment generating function of \(W\), and use it to calculate the expected lifetime.

Short Answer

Expert verified
The expected lifetime of the system is 2000 hours.

Step by step solution

01

Understand the Exponential Distribution

The lifetime of each component is exponentially distributed with a mean of 1000 hours. For an exponential distribution, the parameter \( \lambda \) is the reciprocal of the mean, so \( \lambda = \frac{1}{1000} \).
02

Define System Lifetime

The system functions as long as at least one of the components is functioning. Therefore, the system lifetime \( W \) is the maximum of the lifetimes of the two components.
03

Maximum of Two Exponential Variables

For independent exponential random variables \( X \) and \( Y \) with the same parameter \( \lambda \), the maximum is distributed exponentially with parameter \( \lambda/n \) where \( n \) is the number of variables. Here, since \( n = 2 \), the new parameter \( \lambda' = \frac{1}{1000} / 2 = \frac{1}{2000} \).
04

Moment Generating Function of an Exponential Distribution

The moment generating function (MGF) for an exponential random variable with parameter \( \lambda' \) is \( M_W(t) = \frac{\lambda'}{\lambda' - t} \) for \( t < \lambda' \). Using \( \lambda' = \frac{1}{2000} \), the MGF is \( M_W(t) = \frac{1/2000}{1/2000 - t} \).
05

Calculate the Expected Lifetime using MGF

The expected lifetime, \( E[W] \), of an exponential distribution can be obtained by differentiating the logarithm of its MGF with respect to \( t \) and evaluating at \( t = 0 \). Since \( E[W] = \frac{1}{\lambda'} \), we have \( E[W] = 2000 \).

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Key Concepts

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

Exponential Distribution
The exponential distribution is a commonly used probability distribution in reliability theory and statistics. It is often used to model the time until an event occurs. For instance, the lifetime of components until they fail. One key characteristic of the exponential distribution is its memorylessness, meaning the probability of an event occurring in the future is not dependent on how much time has already elapsed.

The distribution is defined by just one parameter, \( \lambda\\), which is the rate parameter. This parameter is the reciprocal of the mean. In our exercise, each component has a mean lifetime of 1000 hours, so \( \lambda = 1/1000\\).

This simplicity makes the exponential distribution a perfect choice for modeling systems where events happen continuously and independently at a constant average rate.
System Lifetime
System lifetime refers to the period a system can perform its function. In our example, the system is composed of two components in parallel. This setup means the system will keep functioning as long as at least one component is still operational.

The functioning of the system is therefore dependent on the longest-lasting component. Thus, the system lifetime, denoted by \(W\), is determined by the maximum of the lifetimes of the two components. Such a configuration is beneficial as it extends the lifespan of the system overall, taking advantage of the redundancy provided by parallel components.
Independent Random Variables
The concept of independent random variables is crucial in probability theory, especially in this context. Independence means that the lifetime of one component does not influence the lifetime of the other.

Mathematically, two random variables \(X\) and \(Y\) are independent if the joint probability distribution can be expressed as the product of their marginal distributions. In practical terms, the performance of one component doesn't affect or predict the performance of the other. This property simplifies the analysis since one can treat the two variables separately when modeling their behavior.
Expected Lifetime
The expected lifetime or mean is a measure of the central tendency of a probability distribution, representing the average time before the system ceases to function. In the case of our exercise with exponential distributions and parallel components, we use the maximum of their lifetimes.

The maximum of two independent exponentially distributed variables with the same rate parameter, \(n = 2\), results in a new exponential distribution with a reduced parameter, \(\lambda' = \lambda/2 = 1/2000\). Thus, the expected lifetime of the system \((E[W])\) is simply the reciprocal of \(\lambda'\), resulting in \(E[W] = 2000\) hours. This expectation provides important insights, helping predict how long the system will likely perform without failure.

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Most popular questions from this chapter

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