Question 1
Briefly explain the concepts of independence and dependence between two events.
Question 2
Consider the costs of two project items given in Fig. 2.1, with identical discrete probability distributions. Theyare to be linearly added.
Fig. 2.1
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Item I
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Cost ($M)
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Probability
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13.0
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0.150
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16.5
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0.600
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20.0
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0.250
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Item 2
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Cost ($M)
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Probability
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13.0
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0.150
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16.5
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0.600
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20.0
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0.250
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(i) Assume independence between the two items.
1. Calculate the points of the combined probability distribution of total cost of the two items.
2. Calculate mean (expected value), variance and mode (most likely value) of the
consequence of this total cost.
3. Plot the frequency distribution and cumulative probability distribution of the result.
(ii) Assume complete (100%) positive dependence between the costs.
1. Calculate the points of the combined probability distribution of total cost of the two items.
2. Calculate mean (expected value), variance and mode (most likely value) of the consequence of this total cost.
3. Plot the frequency distribution and cumulative probability distribution of the result.
Question 3
(i) Assuming a modified probability distribution as given in Fig. 3.1, in which the cost of the second item is conditionally dependent on the first:
1. Calculate the points of the combined probability distribution of total cost of the two items.
2. Calculate mean (expected value), variance and mode (most likely value) of the consequence of this total cost.
3. Plot the frequency distribution and cumulative probability distribution of the result.
Fig. 3.1
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Risk 1
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Cost ($M)
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Probability
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13.0
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0.150
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16.5
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0.600
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20.0
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0.250
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Risk 2
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Cost ($M)
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Probability
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13.0
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0.350
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16.5
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0.650
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13.0
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0.200
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16.5
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0.550
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20.0
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0.250
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13.0
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0.050
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16.5
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0.800
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20.0
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0.150
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(ii) Comment on the calculations you have made in questions 2 and 3, explaining any key points of similarity or difference between the results (up to 300 words).
(iii) Discuss examples of when, in an engineering or technological project or process,you would expect to find examples of each of these three scenarios (independence between the costs of two items, complete positive dependence between the costs of two items, and conditional dependence of the cost of one item on another) (up to 300 words).
Further Information
Background reading material for this task may be found in Chapman and Ward, Chapter 11. For this question, you can consider the variables as having discrete probabilities (i.e., not continuous).