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Exercise 2. Show that the exponential distribution for time to failure has the forgetfulness property: P(T≥ to +ti T≥ to) = P(T≥t₁) (4) = What is Equation) saying? The probability the device functions for another, say, t₁ 100 hours doesn't depend on how long its been working already. The device isn't subject to wear out. This is why the exponential distribution is a very poor choice for modeling the life of mechanical devices. However, for

Exercise 2. Show that the exponential distribution for time to failure has the forgetfulness property: P(T≥ to +ti T≥ to) = P(T≥t₁) (4) = What is Equation) saying? The probability the device functions for another, say, t₁ 100 hours doesn't depend on how long its been working already. The device isn't subject to wear out. This is why the exponential distribution is a very poor choice for modeling the life of mechanical devices. However, for

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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4 2.1 Some Rehability Theory Recall that an exponential random variable is a non-negative random variable with probability density function is given by Equation: f(t) = Aet; t>0 (1) The exponential distribution arises in many applications. In the theory of queues, (waiting lines), the time between events, e.g. arrivals at a waiting line, is often well modeled by an exponential distribution. In reliability theory, the time between failures for electronic components is often modeled as following an exponential distribution. We will investigate the latter application. Axiomatic Development Let T be the time until failure for a device a transistor, circuit board, or maybe an entire communications system. Proceeding axiomatically, we make the following assumptions: 1. For any two disjoint time intervals, the event that a failure occurs in one interval is independent of the event that a failure occurs in the other time interval. 2. For a time interval of length At, the probability of a failure in the interval is proportional to At if At is small. That is for any t, there is a constant > 0 so that P(t ≤T≤t+At) At, when At is small. Define the reliability function as R(t) = P(T2 t). Then the axioms imply that for At small R(t + At) = R(t)(1-XAt) (2) Why? To still be functioning at time t+ At, there was no failure in the interval (0, t) and no failure in the interval [t,t + At). Rearranging the equation and taking the limit as At →0, this gives dR dt =-AR (3) Hence R(t) = R(0)e-t. One must have R(0) = 1 (Why?); hence F(t) = 1-e- and so f(t) = Ae-M: the exponential distribution. 5: Epic e CAID SELF PAY CODES 58,59,80,82,89 Page < 2 > of 5 57°F Mostly clo BJC210
Transcribed Image Text:4 2.1 Some Rehability Theory Recall that an exponential random variable is a non-negative random variable with probability density function is given by Equation: f(t) = Aet; t>0 (1) The exponential distribution arises in many applications. In the theory of queues, (waiting lines), the time between events, e.g. arrivals at a waiting line, is often well modeled by an exponential distribution. In reliability theory, the time between failures for electronic components is often modeled as following an exponential distribution. We will investigate the latter application. Axiomatic Development Let T be the time until failure for a device a transistor, circuit board, or maybe an entire communications system. Proceeding axiomatically, we make the following assumptions: 1. For any two disjoint time intervals, the event that a failure occurs in one interval is independent of the event that a failure occurs in the other time interval. 2. For a time interval of length At, the probability of a failure in the interval is proportional to At if At is small. That is for any t, there is a constant > 0 so that P(t ≤T≤t+At) At, when At is small. Define the reliability function as R(t) = P(T2 t). Then the axioms imply that for At small R(t + At) = R(t)(1-XAt) (2) Why? To still be functioning at time t+ At, there was no failure in the interval (0, t) and no failure in the interval [t,t + At). Rearranging the equation and taking the limit as At →0, this gives dR dt =-AR (3) Hence R(t) = R(0)e-t. One must have R(0) = 1 (Why?); hence F(t) = 1-e- and so f(t) = Ae-M: the exponential distribution. 5: Epic e CAID SELF PAY CODES 58,59,80,82,89 Page < 2 > of 5 57°F Mostly clo BJC210
Exercise 2. Show that the exponential distribution for time to failure has the forgetfulness property: P(T≥ to +ti|T≥ to) = P(T≥ t₁) (4) What is Equation ) saying? The probability the device functions for another, say, ti 100 hours doesn't depend on how long its been working already. The device isn't subject to wear out. This is why the exponential distribution is a very poor choice for modeling the life of mechanical devices. However, for electronic devices the causal system that leads to failure is often not age related, and the exponential distribution can be a reasonable model for failure times.
Transcribed Image Text:Exercise 2. Show that the exponential distribution for time to failure has the forgetfulness property: P(T≥ to +ti|T≥ to) = P(T≥ t₁) (4) What is Equation ) saying? The probability the device functions for another, say, ti 100 hours doesn't depend on how long its been working already. The device isn't subject to wear out. This is why the exponential distribution is a very poor choice for modeling the life of mechanical devices. However, for electronic devices the causal system that leads to failure is often not age related, and the exponential distribution can be a reasonable model for failure times.
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