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MA 583 Midterm Exam

代考随机过程 Calculate u1. Show all of your work and explain each step. For fullcredit, your final answer should not contain an infinite sum.

Instructions:

• Upload a PDF of your exam solutions before 8:00 AM EDT July 24, 2020.
• Explain all of your steps.
• There is no need to simplify arithmetic (unless statedotherwise).  代考随机过程
• You may refer to the textbook, recorded lectures, homework assign- ments, and lecture notes.
• You may not use any resources other than the textbook, recorded lec- tures, homework assignments, and lecture notes.
• You may not use a calculator or calculator website.
• You may not collaborate with anybody.
• Do not cheat.

1.(20points) Follow these steps to answer the following question about coin flflips.  代考随机过程

• Flipa fair coin repeatedly until you get three heads in a row.
• Eachflip is  Each flip has a probabilityof being heads and a probabilityof being tails.
• Let X_n be a Markov chain whose value records the number of heads in a row.

-If the nth flip is tails then X_n = 0.

-If the nth flip is heads and the (n − 1)th flip is not heads then X_n = 1.  代考随机过程

-If the nth flip is heads and the (n − 1)th flip is heads and the (n − 2)th flip is not heads then X_n = 1.

-If the nth flip is heads and the (n − 1)th flip is heads and the (n − 2)th flip is heads then X_n = 3.

QUESTION: What is the expected total number of tails that you flip before flipping three heads in a row?

(a)Write down a transition probability matrix P for this Markov Show all of your work and explain eachstep.

(b)Let T = min{n ≥ 0 : X_n= 3}. Find a function g such that

counts the total number of times tails is flipped. Explain your answer.

(c)Let w_i:= E(Z|X₀ = i) and set up a system of linear system of equations for w₀, w₁, and w₂ and SOLVE FOR w₀, w₁, and w₂. Show all of your work and explain each step.

(d)Calculate E(Z). Show all of your work and explain each step.

2.(20points) Let X_n be a Markov chain with transition probability matrix  代考随机过程

Assume that the initial distribution of X₀ is

P(X₀ = 0) = .2 P(X₀ = 1) = .4 P(X₀ = 2) = .4.

(a)Calculate P(X₁= 0), P(X₁ = 1), and P(X₁ = 2). SIMPLIFY YOUR ARITHMETIC! Show all of your work and explain each step.

(b)Calculate thelimits

and

WRITE YOUR ANSWER EXACTLY. DO NOT GIVE A DEC-IMAL APPROXIMATION. Show all of your work and explain each step.

3.LetX_n be a branching process with descendents having i.d. geometric distributions

A branching process means that

Let un := P(Xn = 0|X0 = 1).

(a)Calculate u₁. Show all of your work and explain each step. For fullcredit, your final answer should not contain an infinite sum.

(b)Calculate u₂. Show all of your work and explain each step. For fullcredit, your final answer should not contain an infinite sum.

(c)Calculate lim u_n. Your answer should be Show all of your work and explain each step.

(d)Whatis the probability that X_n eventually equals zero if the initial population size is X₀ = 4? Show all of your work and explain each step.

4.(20points) Let X_n be a Markov chain with transition matrix

(a)Identify the communicating classes of this Markov Show allof your work and explain each step.

(b)Identifythe period of state  Show all of your work and explain each step.

(c)Verify that thedistribution

x = (.2 .5 .3 0 0)

solves x = xP .

(d)Verify that the distribution

solves y = yP . Show your work.

(e)Explain why thelimit

does not exist. Show your work.

(f) Explain why the limit

Show your work.

5.(20points) Write down a probability transition matrix for the following situation.  代考随机过程

• Start by rolling one
• If the result of the previous roll is even then roll two dice on the next turn.
• If the result of the previous roll was odd, roll one die on the next turn.
• If two dice are rolled and their sum is even, roll two dice on the next turn.  代考随机过程
• If two dice are rolled and their sum is odd, roll one die on the next turn.
• Thegame ends when you roll a sum of 7 or 12.

• Writedown a one-step probability transition matrix for a Markov chain that can describe this  Make sure to explain clearly what each state represents. Show all of your work explaining how you calculated the transition probabilities.
• Writedown a system of linear equations that you can use to figure out the probability of the game ending with a sum of 12. SOLVE THE  Make sure to label all of your variables clearly and identify which variable answers the question.

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