Stochastic Processes : Cumulative Distribution Function, Renewal Function and Poisson Process
Not what you're looking for?
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 andfortx.
(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)
Purchase this Solution
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.
Purchase this Solution
Free BrainMass Quizzes
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Probability Quiz
Some questions on probability
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.