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We have seen that if E(X,) = E(X,) = ... = E(X,) = H, then E(X, + ... + X,) = nµ. In some applications, the number of X's under consideration is not a fixed number n but instead is an rv N. For example, let N = the number of components that are brought into a repair shop on a particular day, and let X, denote the repair shop time for the ith component. Then the total repair time is X, + X, + ... + XN, the sum of a random number of random variables. When N is independent of the X's, it can be shown that E(X, + ... + XN) = E(N) • µ. (a) If the expected number of components brought in on a particularly day is 20 and expected repair time for a randomly submitted component is 30 min, what is the expected total repair time, in minutes, for components submitted on any particular day? min (b) Suppose components of a certain type come in for repair according to a Poisson process with a rate of 5 per hour. The expected number of defects per component is 4.5. What is the expected value of the total number of defects on components submitted for repair during a 6-hour period? Be sure to indicate how your answer follows from the general result just given. For this process we would expect components to come in for repair during a 6-hour period. This number is E(N) vy using the above notational scheme. Similarly, we would expect to see defects on each repair. This number is v in the above notational scheme. So, we would expect to see defects over a 6-hour period time period.

We have seen that if E(X,) = E(X,) = ... = E(X,) = H, then E(X, + ... + X,) = nµ. In some applications, the number of X's under consideration is not a fixed number n but instead is an rv N. For example, let N = the number of components that are brought into a repair shop on a particular day, and let X, denote the repair shop time for the ith component. Then the total repair time is X, + X, + ... + XN, the sum of a random number of random variables. When N is independent of the X's, it can be shown that E(X, + ... + XN) = E(N) • µ. (a) If the expected number of components brought in on a particularly day is 20 and expected repair time for a randomly submitted component is 30 min, what is the expected total repair time, in minutes, for components submitted on any particular day? min (b) Suppose components of a certain type come in for repair according to a Poisson process with a rate of 5 per hour. The expected number of defects per component is 4.5. What is the expected value of the total number of defects on components submitted for repair during a 6-hour period? Be sure to indicate how your answer follows from the general result just given. For this process we would expect components to come in for repair during a 6-hour period. This number is E(N) vy using the above notational scheme. Similarly, we would expect to see defects on each repair. This number is v in the above notational scheme. So, we would expect to see defects over a 6-hour period time period.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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We have seen that if E(X,) = E(X,) N = the number of components that are brought into a repair shop on a particular day, and let X; denote the repair shop time for the ith component. Then the total repair time is X, + X, + ... random number of random variables. When N is independent of the X;'s, it can be shown that = ... = E(X,) = µ, then E(X, + ... + X,) = nụ. In some applications, the number of X's under consideration is not a fixed number n but instead is an rv N. For example, let + XN, the sum of a E(X, + ... + X ) = E(N) • µ. (a) If the expected number of components brought in on a particularly day is 20 and expected repair time for a randomly submitted component is 30 min, what is the expected total repair time, in minutes, for components submitted on any particular day? min (b) Suppose components of a certain type come in for repair according to a Poisson process with a rate of 5 per hour. The expected number of defects per component is 4.5. What is the expected value of the total number of defects on components submitted for repair during a 6-hour period? Be sure to indicate how your answer follows from the general result just given. For this process we would expect components to come in for repair during a 6-hour period. This number is E(N) using the above notational scheme. Similarly, we would expect to see defects on each repair. This number is u in the above notational scheme. So, we would expect to see defects over a 6-hour period time period.
Transcribed Image Text:We have seen that if E(X,) = E(X,) N = the number of components that are brought into a repair shop on a particular day, and let X; denote the repair shop time for the ith component. Then the total repair time is X, + X, + ... random number of random variables. When N is independent of the X;'s, it can be shown that = ... = E(X,) = µ, then E(X, + ... + X,) = nụ. In some applications, the number of X's under consideration is not a fixed number n but instead is an rv N. For example, let + XN, the sum of a E(X, + ... + X ) = E(N) • µ. (a) If the expected number of components brought in on a particularly day is 20 and expected repair time for a randomly submitted component is 30 min, what is the expected total repair time, in minutes, for components submitted on any particular day? min (b) Suppose components of a certain type come in for repair according to a Poisson process with a rate of 5 per hour. The expected number of defects per component is 4.5. What is the expected value of the total number of defects on components submitted for repair during a 6-hour period? Be sure to indicate how your answer follows from the general result just given. For this process we would expect components to come in for repair during a 6-hour period. This number is E(N) using the above notational scheme. Similarly, we would expect to see defects on each repair. This number is u in the above notational scheme. So, we would expect to see defects over a 6-hour period time period.
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