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
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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.
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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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