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QUESTION ONE

• A unit of electrical equipment is subject to failure. The probability distribution of its age at failure is:

Age at failure (weeks)	1	2	3	4	5
Probability	0	0.2	0.4	0.3	0.1

Initially 10,000 new units are installed and any unit which fails is replaced by a new unit at the end of the week in which it fails

• Calculate the expected number of units to be replaced in each of weeks 1 to 7. What rate of failures can be expected in the long run?
• Replacement of individual units on failure costs $5 each. An alternative policy is to replace all units after a fixed number of weeks at a cost of $30 and to replace any units failing before the replacement week at individual cost of $5 each.
• Should this group replacement policy be adopted?
• If so, after how many weeks should all units be replaced?

QUESTION TWO

• A company wishes to design an electronic device consisting of three main components. The three components are arranged in series so that the failure of one of the components will result in the failure of whole device. Therefore it is decided that the reliability (probability of failure) of the device can be increased by installing parallel units on each component. Each component may be installed at most 3 parallel units. The total capital (in million Kenyan shillings) available is 10 million. The following data is available.

Mi	i = 1	i	i
r1	c1	r2	c2	r3	c3
1	0.5	2	0.7	3	0.6	1
2	0.7	4	0.8	5	0.8	2
3	0.3	5	0.9	6	0.9	3

Where M_i is the number of parallel units placed with the ith component,

r_i is the reliability of the component

c_i is the cost of the ith component

Determine m which will maximize the total reliability of the system without exceeding the given capital.

QUESTION THREE

• Giving reasons, classify the states of the following Markov chain

• A man either drives his car or takes a train to work each day. Suppose he never takes the train two days in a row, but if he drives to work, then the next day he is just likely to drive again as he is to take the train.
• Find the transition matrix
• Determine the probability of the changes from going by train to driving exactly in four days
• A house wife buys three kinds of cereals: A, B, C. she never buys the same cereal on successive week. If she buys cereal A, then the next week she buys cereal B. However, if she buys B or C, then the next day he is twice as likely to sell in city A as in the other city. In the long run, how often does he sell in each of the cities?

QUESTION ONE a. A unit of electrical equipment is subject to failure. The probability distribution of its age at failure is: Age at failure (weeks) Probability 1 2 3 4 5 0 0.2 0.4 0.3 0.1 Initially 10,000 new units are installed and any unit which fails is replaced by a new unit at the end of the week in which it fails i. ii. Calculate the expected number of units to be replaced in each of weeks 1 to 7. What rate of failures can be expected in the long run? Replacement of individual units on failure costs $5 each. An alternative policy is to replace all units after a fixed number of weeks at a cost of $30 and to replace any units failing before the replacement week at individual cost of $5 each. a. Should this group replacement policy be adopted? b. If so, after how many weeks should all units be replaced? QUESTION TWO i. A company wishes to design an electronic device consisting of three main components. The three components are arranged in series so that the failure of one of the components will result in the failure of whole device. Therefore it is decided that the reliability (probability of failure) of the device can be increased by installing parallel units on each component. Each component may be installed at most 3 parallel units. The total capital (in million Kenyan shillings) available is 10 million. The following data is available. Mi 1 2 3 r1 0.5 0.7 0.3 i=1 c1 2 4 5 r2 0.7 0.8 0.9 i=2 c2 3 5 6 r3 0.6 0.8 0.9 i=3 c3 1 2 3 Where Mi is the number of parallel units placed with the ith component, ri is the reliability of the component ci is the cost of the ith component Determine m which will maximize the total reliability of the system without exceeding the given capital. QUESTION THREE a. Giving reasons, classify the states of the following Markov chain 0 0 0 1 0 3 / 4 1/ 8 1/ 8 f (n) 1 / 8 0 3 / 4 1 / 8 0 0 1 / 6 5 / 6 b. A man either drives his car or takes a train to work each day. Suppose he never takes the train two days in a row, but if he drives to work, then the next day he is just likely to drive again as he is to take the train. i. Find the transition matrix ii. Determine the probability of the changes from going by train to driving exactly in four days c. A house wife buys three kinds of cereals: A, B, C. she never buys the same cereal on successive week. If she buys cereal A, then the next week she buys cereal B. However, if she buys B or C, then the next day he is twice as likely to sell in city A as in the other city. In the long run, how often does he sell in each of the cities