闪电代写 -代写CS作业_CS代写_Finance代写_Economic代写_Statistics代写_代码代做_IT代写_加急帮助

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Homework #6

(Type A: due Friday, February 17, at midnight)

(Type B: due Monday, February 20, at 3:30 PM)

Type A Problems: You may consult the instructor, the TAs, or other Math 180B students. However, you may not share answers, and you must write your ﬁnal solutions independently and acknowledge any help. Also, you may not look for help online or discuss these problems on social media sites such as Discord.

1. Suppose a gambler has $30 and wants to increase her fortune to $50. She will continue gambling until she either reaches her goal of $50 or until she runs out of money. Each time she makes a bet, she either wins or loses the amount of her bet.

(a) Suppose she bets $1 each time and wins each bet with probability 0.5. What is the probability that she gets to $50 before running out of money?

(b) Suppose she bets $10 each time and wins each bet with probability 0.5. What is the probability that she gets to $50 before running out of money?

(c) Suppose she bets $1 each time and wins each bet with probability 0.4. What is the probability that she gets to $50 before running out of money?

(d) Suppose she bets $10 each time and wins each bet with probability 0.4. What is the probability that she gets to $50 before running out of money?

2. Let (Xn) be a Markov chain with state space S = e1, 2, 3, 4{ and transition matrix

┌ ┐

P = ''' .2 .3 0 .5 ''' .

(a) If X0 = 2, what is the probability that this Markov chain reaches 1 before 4? (b) If X0 = 2, on average how long does it take for the Markov chain to reach 1 or 4?

3. Consider the Ehrenfest urn, in which there are 5 balls and two urns. At each time step, we pick a ball uniformly at random and move it to the other urn. If we begin with 1 ball in the ﬁrst urn and 4 balls in the second urn, what is the probability that the ﬁrst urn becomes empty before the second urn is ever empty?

4. Consider the following three-stage manufacturing process. An item begins at Stage 1. An item at Stage 1 moves next to Stage 2 with probability 0.95 and has to be discarded with probability 0.05. An item that gets to Stage 2 moves to Stage 3 with probability 0.9 and is sent back to Stage 1 with probability 0.1. An item that gets to Stage 3 is successfully manufactured with probability 0.98 and has to be discarded with probability 0.02. What is the probability that eventually the item is successfully manufactured?

5. Consider an urn that has three balls, each of which is either black or white. At each time step, we pick two balls out of the urn at random and replace them both with black balls.

(a) Model the number of black balls in the urn as a Markov chain. Give the state space and transition probabilities.

(b) If initially all three balls are white, how many steps does it take on average until all balls in the urn are black?

6. Consider the random walk on the hexagon. At each step, the random walker moves to one of the two neighboring vertices with equal probabilities. For example, from 1, the walker

can go to 2 or 6.

3 4

↘│ │╱

1 6

For j = 2, 3, 4, 5, 6, ﬁnd the expected number of steps that it takes for the random walker to reach 1 starting from j .

Hint: You can simplify the algebra by taking advantage of symmetries.

7. Suppose (Xn) is a Markov chain with state space S = e0, 1, 2, . . . {. Suppose the tran- sition probabilities are given by p(i, 0) = 1/(i + 1) and p(i, i + 1) = i/(i + 1) for i _ 1 and p(0, 0) = 1. Suppose X0 = 1. Let m be a positive integer. What is the probability that the Markov chain (Xn) reaches the state m before reaching 0?

8. Suppose that the number of cars that a salesperson sells in a day has a Poisson distribution with mean 1.6. On average, how many days will go by before there are 4 consecutive days in which no cars are sold?

9. Let (Xn) be a Markov chain with state space S = e0, 1, 2, 3, 4{ and transition probabili- ties p(0, 0) = p(4, 4) = 1 and p(i, i _ 1) = p(i, i + 1) = 1/2 for i > e1, 2, 3{. That is, (Xn) is the simple random walk, absorbed when it reaches 0 or 4. Let N be the number of times the Markov chain visits the state 2, before being absorbed at 0 or 4. Let f (i) = Ei[N], which is the expected number of visits to 2 when X0 = i. Calculate f (i) for i > S .

Hint: use ﬁrst-step conditioning to derive a system of equations for the numbers f (i).

Selected Answers:

1a) 0.6, 1b) 0.6, 1c) 0.0003, 1d) 0.36

2a) .815, 2b) 1.605

3) 5/8

Type B Problems: You must work completely alone on these problems.

1. A gambler starts out with $1 and wishes to increase his fortune to $5. Each time the gambler makes a bet, he wins the amount of the bet with probability 1/2 and loses the amount of the bet with probability 1/2. The gambler decides to follow a strategy called “bold play” . If he has $2 or less, he bets everything he has. If he has $3, then he bets $2, and if he has $4, then he bets $1, so that he gets to $5 if he wins. He keeps placing bets until either his fortune reaches $5 or he loses all of his money.

(a) Model this process by a Markov chain. Give the state space and transition matrix. (b) Find the probability that the gambler succeeds in accumulating $5 before going broke.

2. Let (Xn) be a discrete-time Markov chain with state space S = e1, 2{ and transition matrix