﻿ Stochastic Modeling代考 Midterm Exam代写 – 天才代写

# Stochastic Modeling代考 Midterm Exam代写

2023-05-31 15:07 星期三 所属： 考试助攻 浏览：70 ## MA 583 Midterm Exam

Stochastic Modeling代考 Use fifirst-step analysis, conditioning on the possible values of X1,to show that the generating function ψn(s) solves the recursion

### Instructions:

• Submit your solutions as a single PDF on learn.bu.edu.
• Solutions are due before 8:00AM on Thursday March 11, 2021.
• Explain all of your steps. YOU MUST WRITE AT LEAST ONE SEN TENCE OF EXPLANATION IN EACH STEP OF EACH PROBLEM.
• Unless stated otherwise, you need to solve any linear algebra problems by hand and you must show all of your work.
• You do not need to use a calculator. Make sure that you write any arithmetic onto your exam solutions. I will grade the arithmetic for mula as the fifinal answer, not any decimal approximation.  Stochastic Modeling代考
• You may use your notes, the textbook, and anything on the course webpage at learn.bu.edu.
• You may not use any other sources including other books or anything on the internet.
• You may not collaborate with anybody.
• Do not cheat.
• Some summation formulas are on the back page.Salins

### 1.(25 points) Let Xn be a Markov chain on the states {0, 1, 2, 3, 4} with the one-step transition matrix and initial distribution

Let T = min{n : Xn 2} be the fifirst time that the Markov chain hits one of the absorbing states.

(a) Calculate P(XT = 4|X0 = 0). Show all of your work and explain each step.

(b) Calculate P(XT = 4). Show all of your work and explain each step.

### 2.Let ξ be a non-negative integer-valued random variable with probability generating function   Stochastic Modeling代考 (a) Calculate the pmf for ξ. That means calculate P(ξ = k) for every integer k N. Show all of your work and explain each step.

(b) Calculate the expectation E[ξ]. Show all of your work and explain each step.

(c) Show that there exists a random variable X with generating func-tion Hint: Random sums of random variables.

### 3.Let

be a transition probability matrix for a MArkov chain on the states {0, 1, 2, 3, 4}.

(a) Identify the communicating classes of the matrix. Show all of your work and explain each step.

(b) Show that µ = is an invariant distribution of P. Show all of your work and explain each step.

(c) Find an invariant distribution of the form ν =(0  0  0  a  b).Show all of your work and explain each step.

(d) Show that is an invariant distribution of P  where a and b are the numbers from part (c). Show all of your work and explain each step.

### 4.Let Xn be a branching process where the descendant distribution has generating function φ(s) and the initial population is X0 = 1. Let  Stochastic Modeling代考 be the cumulative population size up to time n. Let ψn = be the generating function of Sn.

(a) Let µ = φ’ (1). Calculate E(Zn) for any n. Show all of your work and explain each step. For full credit, your answer should not have a summation in it.

(b) Use fifirst-step analysis, conditioning on the possible values of X1,to show that the generating function ψn(s) solves the recursion

ψn(s) = (ψn1(s)).

Show all of your work and explain each step.

(c) Assume that the descendant distribution has generating function In this case, the limit  