please do give me detailed explanations and solution for all problems present in document. below are the problems.

1. A machine is inspected at the end of every hour. It is found to be either working or failed. If the machine is found to be working, the probability of its remaining up for the next hour is 0.90. If is found to be failed, the machine is repaired which may take more than one hour. Whenever the machine is failed (regardless of how long it has been down), the probability of its still being down one hour later is 0.35.

Part A: Construct a one-step transition matrix for this Markov chain.

Part B: Determine the mean first passage time from state i to j for all i and j.

Part C: What is the mean number of hours until the first failure if it is currently working?

2. A handbag company specializes in a luxury-type handbag. The sales of this handbag fluctuate between two levels- low and high- depending upon two factors: (1) whether they advertise and (2) the advertising and marketing of new products being done by competitors. The second factor is out of the company's control, but it is trying to determine what its own advertising policy should be. For example, the marketing manager's proposal is to advertise when sales are low, but not to advertise when sales are high. Advertising in any quarter of a year has its primary impact on sales in the following quarter. Therefore, at the beginning of each quarter, the needed information is available to forecast accurately whether sales will be low or high that quarter and to decide whether to advertise that quarter. The cost of advertising is $1 million for each quarter of a year in which it is done. When advertising is done during a quarter, the probability of having high sales the next quarter is 1/2 or 3/4, depending on whether the current quarter's sales are low or high. These probabilities go down to 1/4 or 1/2 when advertising is not done during the current quarter. The company's quarterly profits (excluding advertising costs) are $4 million when sales are high but only $2 million when sales are low. (Hereafter, use units of millions of dollars.)

Part A: Construct a one-step transition matrix for each of the following advertising strategies: (i) never advertise, (ii) always advertise, and (iii) follow the marketing manager's proposal.

Part B: Determine the steady-state probabilities for each of the three cases in part A

Part C: Find the long-run expected average profit (including a deduction for advertising costs) per quarter for each of the three advertising strategies in part A.

Part D: Which of these strategies is best according to this measure of performance?

Part E: Given the handbag company example in question 2, list and explain two specific conditions that are possible that could cause the Markov assumption to be invalid for that model.

Part F: Given the handbag company example in question 2, list and explain two specific conditions that are possible that could cause the stationary assumption to be invalid for that model.

3. Given the following one-step transition matrix of a Markov chain, determine:

Part A: The classes of the Markov chain.

Part B: Indicate whether they are recurrent.

1/4 |
3/4 |
0 |
0 |
0 |

3/4 |
1/4 |
0 |
0 |
0 |

1/3 |
1/3 |
1/3 |
0 |
0 |

0 |
0 |
0 |
3/4 |
1/4 |

0 |
0 |
0 |
1/4 |
3/4 |

Transition diagram that identifies the classes and transient/recurrent status.

Subject | Mathematics |

Due By (Pacific Time) | 10/30/2014 12:00 am |

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