/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 If \(A(t)\) and \(Y(t)\) are, re... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 31

If \(A(t)\) and \(Y(t)\) are, respectively, the age and the excess at time \(t\) of a renewal process having an interarrival distribution \(F\), calculate $$ P\\{Y(t)>x \mid A(t)=s\\} $$

Short Answer

Expert verified
The short answer for the given problem is: $$ P\{Y(t) > x \mid A(t) = s\} = \frac{P\{X_{N(t)+1} > s + x\}}{P\{X_{N(t)+1} > s\}} $$ where \(X_{N(t)+1}\) is the remaining time till the next event in the renewal process.

Step by step solution

01

Understand the problem

We need to find $$ P\{Y(t) > x \mid A(t) = s\} $$ where \(A(t)\) and \(Y(t)\) are the age and excess of a renewal process at time \(t\), which has interarrival distribution \(F\).
02

Use the definition of conditional probability

By definition, the conditional probability is given by $$ P\{Y(t) > x \mid A(t) = s\} = \frac{P\{Y(t) > x \cap A(t) = s\}}{P\{A(t) = s\}} $$
03

Determine the joint probability of \(Y(t) > x\) and \(A(t) = s\)

\(A(t) = s\) means that the renewal process has just completed its \(s\)th arrival at time \(t\). For \(Y(t) > x\), we have the remaining time until the next arrival is greater than \(x\) with respect to that. So the joint probability is $$ P\{Y(t) > x \cap A(t) = s\} = P\{A(t) = s, N(t+s) - N(t) = 0\} $$ where \(N(t)\) is the number of arrivals in the process up to time \(t\).
04

Determine the probability of \(A(t) = s\)

The probability of \(A(t) = s\) can be determined by examining the time since the last event. Specifically, it is the probability of having an arrival at time \(t-s\) and no arrivals within the interval \([t-s,t)\): $$ P\{A(t) = s\} = P\{N(t-s) - N(t) = -1\} $$
05

Use renewal process properties

We know the renewal process has interarrival distribution \(F\). We can rewrite the joint probability as $$ P\{Y(t) > x \cap A(t) = s\} = P\{A(t) = s, N(t+s) - N(t) = 0\} = P\{X_{N(t)+1} > s+x\} $$ and we can rewrite the probability of \(A(t) = s\) as $$ P\{A(t) = s\} = P\{N(t-s) - N(t) = -1\} = P\{X_{N(t)+1} > s\} $$
06

Calculate the conditional probability

Now we have everything we need to calculate the conditional probability. Substitute the obtained results back into the equation obtained in Step 2: $$ P\{Y(t) > x \mid A(t) = s\} = \frac{P\{X_{N(t)+1} > s + x\}}{P\{X_{N(t)+1} > s\}} $$ Note that the conditional probability does not depend on \(t\).

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Most popular questions from this chapter

If the mean-value function of the renewal process \(\\{N(t), t \geqslant 0\\}\) is given by \(m(t)=\) \(t / 2, t \geqslant 0\), what is \(P[N(5)=0\\} ?\)

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