A component of a computer has an active life, measured in discrete units, that is a random variableTwhere P(T=1)=0.4,P(T=2)=0.
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A component of a computer

has an active life, measured in discrete units, that is a random variable T where

P(T =1)=0.4, P(T=2)=0.6.

Suppose one starts with a fresh component and each component is replaced by a new component

upon failure.

Define a Markov chain Xn, n=0, 1, . . ., by letting Xn be the age of the component in service at time n. The lifetimes of consecutive components are independently distributed. We also take the convention that Xn=0 at the time of failure. The state space of Xn is {0, 1}.

As an example, if the first component works for 2 periods and the second component works for 1 periods, then we have

X0 =0, X1 =1, X2 =0, X3 =0, ....

(1). Determine the long run probability that a failure occurs in a single period.

Hint: this is related to the limiting distribution of Xn.

(2). What is the expected time between failures?

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