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# Stochastic Processes : Cumulative Distribution Function, Renewal Function and Poisson Process

3. Let {N(t)}>0 be a renewal process for which the interarrival times {T1,T2,. . .} have cumulative distribution function F. Recall that the renewal function m(t) is given by m(t) E[N(t)j.
(a) Find E[N(t)Ti=x] fort<x andfortx.
(b) Find E[N(t)] E[E[N(t) T1]] to prove that m(t) satisfies the renewal equation
m(t) F(t) + f m(t
(c) Show that the renewal function m(t) for a Poisson process with rate A> 0 satisfies the renewal equation.

4. Let {N(t)}>0 be a renewal process for which the interarrival times have cumulative distribution function F and the waiting times are denoted by {W1, W2,. . .}. Define the age of the process {A(t)}>0 by A(t) t
the process {R(t)}>0 by R(t) = WNQ)+1
(a) Find P {R(t)  x A(t) = s}.
(b) Find E [R(t) AQ) s].
(c) Find the answers to parts (a) and (b) in the special case when the interarrival times are exponentially distributed with mean 1/A.

Please see the attached file for the fully formatted problems.

In problems 3 and 4, F is presumably any cdf (no particular distribution was specified).

Also, in problem 4, I the renewal process is continuous (or so I assumed); it may or may not be ergodic.

PPS This is remuneration for problems 3 and 4

(See attached file for full problem description)

#### Solution Summary

Cumulative Distribution Function, Renewal Function and Poisson Process are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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