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)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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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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