CS计算机代考程序代写 chain Salins MA 583 Midterm Exam July 23, 2020 Instructions:
Salins MA 583 Midterm Exam July 23, 2020 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 stated otherwise).
• 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.
Salins MA 583 Midterm Exam July 23, 2020 1. (20 points) Follow these steps to answer the following question about
coin flips.
• Flip a fair coin repeatedly until you get three heads in a row.
• Each flip is independent. Each flip has a probability 1 of being
heads and a probability 1 of being tails. 2
2
• Let Xn be a Markov chain whose value records the number of heads in a row.
– IfthenthflipistailsthenXn =0.
– If the nth flip is heads and the (n − 1)th flip is not heads then
Xn = 1.
– If the nth flip is heads and the (n−1)th flip is heads and the
(n−2)thflipisnotheadsthenXn =2.
– If the nth flip is heads and the (n−1)th flip is heads and the
(n−2)thflipisheadsthenXn =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 chain. Show all of your work and explain each step.
(b) Let T = min{n ≥ 0 : Xn = 3}. Find a function g such that
T
Z = g(Xk)
k=0
counts the total number of times tails is flipped. Explain your
answer.
(c) Let wi := 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.
Salins MA 583 Midterm Exam July 23, 2020 2. (20 points) Let Xn be a Markov chain with transition probability matrix
.2 .5 .3 P=0 .8 .2.
.3 0 .7 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 the limits
and
lim P(Xn = 0), n→∞
lim P(Xn = 1), n→∞
lim P(Xn = 2). n→∞
WRITE YOUR ANSWER EXACTLY. DO NOT GIVE A DEC- IMAL APPROXIMATION. Show all of your work and explain each step.
Salins MA 583 Midterm Exam July 23, 2020 3. Let Xn be a branching process with descendents having i.i.d. geometric
distributions
(n) 1 2i P(ξk =i)=3 3 .
A branching process means that
Xn
Xn+1 = ξ(n+1).
k k=1
Let un := P(Xn = 0|X0 = 1).
(a) Calculate u1. Show all of your work and explain each step. For
full credit, your final answer should not contain an infinite sum.
(b) Calculate u2. Show all of your work and explain each step. For full credit, your final answer should not contain an infinite sum.
(c) Calculate lim un. Your answer should be exact. Show all of your n→∞
work and explain each step.
(d) What is the probability that Xn eventually equals zero if the initial population size is X0 = 4? Show all of your work and explain each step.
Salins MA 583 Midterm Exam July 23, 2020 4. (20 points) Let Xn be a Markov chain with transition matrix
0 1 0 0 0
.4 0 .6 0 0 P=0 1 0 0 0.
0 0 0 . 4 . 6 0 0 0 .2 .8
(a) Identify the communicating classes of this Markov chain. Show all of your work and explain each step.
(b) Identify the period of state 0. Show all of your work and explain each step.
(c) Verify that the distribution
x=.2 .5 .3 0 0
solves x = xP.
(d) Verify that the distribution
y=0001 3 44
solves y = yP . Show your work.
(e) Explain why the limit
lim P(Xn = 0|X0 = 0) n→∞
does not exist. Show your work.
(f) Explain why the limit
lim P(Xn =4|X0 =4)= 3. n→∞ 4
Show your work.
Salins MA 583 Midterm Exam July 23, 2020 5. (20 points) Write down a probability transition matrix for the following
situation.
• Start by rolling one die.
• 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.
• If two dice are rolled and their sum is odd, roll one die on the next turn.
• Thegameendswhenyourollasumof7or12.
(a) Write down a one-step probability transition matrix for a Markov chain that can describe this situation. Make sure to explain clearly what each state represents. Show all of your work explaining how you calculated the transition probabilities.
(b) Write down 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 EQUATIONS. Make sure to label all of your variables clearly and identify which variable answers the question.
