answer all questions with explanation

9. A project is described by the simple activity network shown below, consisting of directed branches representing activities and nodes representing the beginning/termination of activities. The duration in days of two activities A, B, and C in the construction project are denoted as TA, TB, and TC, whose probability mass functions (pmf's) are given below. Assume that TA, TB, and TC are statistically independent. D = 1 2 3 4 5 6 7 Pr{TA=D} 18.5% 39.3% 42.2% Pr{TB=D} 4.5% 5.5% 22.7% 67.3% Pr{TC=D} 10.5% 14.4% 35.6% 7.5% 32% Activities A and C are scheduled to start simultaneously on day 0 [Start]. Node (End) represents the completion of both activities C and B, and the completion of the project. Activity B can start only when A is completed. For example, if (TA, TB, To) = (2, 1, 3), the whole project is completed in T=3 days; and the probability of this happening is (39.3%) (4.5%) (10.5%) = 0.186%. In addition, the whole project is completed in T=3 days if (TA, TB, Tc) = { (1, 2, 3), (1, 1, 3) or (2, 1, 3) } and the probability of this happening is 0.3799% (why?). (a) Verify that Pr(T = 4) = 2.58%. Hint: Nine combinations of (TA, TB, TC) results in T=4. Show your calculations in detail. (b) Compute Pr(T = 6). (c) Determine and plot the pmf of the time T required to complete the project. (d) Determine the most likely time required to complete the project. (e) Compute the mean and the variance of T. (f) Find the smallest value of a such that Pr(T 0.90