Question 6: A contractor estimates the probabilities for the number of days to complete a certain type of construction project as follows:

Time (days) | 1 | 2 | 3 | 4 |

Probability | c | 0.25 | c | c/2 |

where c is a constant, c > 0. It takes at least 1 day and at most 4 days to complete a project of this type.

Let X be a random variable that indicates a number of days to complete a project. Find a constant c so that X has a proper pdf and answer the following questions using the number you found.

What is the probability that a randomly chosen project takes at least 3 days to complete?

Find the expected time to complete the project? Find the standard deviation of time required to complete the project?

The contractor’s project cost is made up of two parts – a fixed cost of $30,000, plus $2,000 for each day taken to complete the project. Find the expected value and standard deviation of total project cost.

If three projects are undertaken, what is the probability that none of them will take more than two days to complete, assuming independence of individual project completion times?