D) The Program Evaluation and Review Technique (PERT) was developed by Booz-Allen Hamilton, Lockheed Martin and the US Navy for the Navy’s Polaris project in 1958. Although the Polaris project was completed before its estimated completion date and under budget, we are not certain that this can be attributed solely to PERT, but PERT was used more widely on future projects after the success of the Polaris project. PERT became popular around the same time computers were progressing from the mainframe to mini-computers. During the evolution of computer technology, advanced programs were developed to provide further probabilistic estimates via simulations (Monte Carlo Analysis).

B)PERT assumes the Beta probability distribution to calculate the expected time of an activity within a network. PERT requires that for each activity, three duration estimates are needed (optimistic, most likely, pessimistic). This distribution is used for PERT since there was no set basis for selecting a specific distribution and was accepted in order to derive an equation. The Beta distribution is also used to calculate the variance and standard deviation of each task and ultimately the entire project via the critical path.

C)The Beta distribution constructs a smooth curve which places more emphasis on values around the most likely value while taking into account the pessimistic and optimistic values. The expected time calculated ideally would be close to the most likely time. Another distribution we could use other than Beta is the triangle distribution, which places even more emphasis on the most likely value than the Beta distribution.

E)The Central Limit Theorem (CLT) states that when using random samples from a population, the sampling distribution of the sample mean can be approximated by a normal distribution as the sample size becomes large (normally sample size of 30 or more). The CLT permits one to add the sum of the random sample means to approximate the population mean as well as add the random sample variances to approximate the population variance. PERT uses the CLT to sum the expected activity times and variances in order to obtain an expected project time and project variance (using the critical path). The square root of the project variance provides us with the total projects standard deviation which can be used to calculate the probability of a project finishing on a specified date.

I) PERT uses the following equation to approximate the standard deviation of a given activity: (Pessimistic Time – Optimistic Time)/6. PERT uses the following equation to approximate the expected time of an activity: (Optimistic Time + 4(Most likely Time) + Pessimistic Time) / 6. In general, the most likely, pessimistic and optimistic times are estimated times that are guessed or based on past history (if this data exists). The probabilities used in the equations for standard deviation and expected times are the following: optimistic time = 1/6 (@ 16.7%), pessimistic time = 1/6 (@ 16.7%), most likely time = 4/6 (@ 66.7%). Per use of the above equations in the beta distribution, different probabilities can not be used for the PERT calculation, however different probabilities should be considered because each activity will not necessarily follow the beta distribution.

J)The Beta distribution is very sensitive to the three-point estimates used, especially when the pessimistic and optimistic time estimates (outside parameters) are changed. We compared the shape of the Beta Distribution of two sets of numbers below with approximately the same mean. The beta distribution below represents the distribution for the following durations in days: 3,4,7 (Optimistic, Most Likely , Pessimistic) which equate to an expected value of 4.3 days.

The resulting distribution is right-skewed. The beta distribution below is based upon the following durations: 1-4-8 (O,M,P) which equate to an expected value...