Problem $2-$ The following table shows the tasks, durations, and predecessor relationships for an urban expressway repair. Draw the precedence network for the project and answer the questions that follow Please assume the Beta distribution for all activity times.

$\backslash$table$[[$Task$,\backslash$table$[[$Immediate$],[$Predecessor$($s$)]],$Time Estimates $($In Weeks$)],[$Optimistic$,$Most Lkely,Pessimistic$],[$A$,$None,$5,6,19],[$B$,$A$,5,6,7],[$C$,$A$,3,4,5],[$D$,$B$,1,3,5],[$E$,$B$,3,4,17],[$G$,$C$,2,4,6],[$H$,$E$,$ F$,3,4,4]]$

a$)$ Draw the network diagram.

b$)$ Calculate the expected duration and variance of each activity.

c$)$ What are the ES $($earliest start$)$ and the LS $($latest start$)$ for each task based on the expected task times?

d$)$ What is the probability of completion of the project before week $17($using the critical path$)$ based on the expected task times?

e$)$ What is the probability that the project will take more than $20$ weeks to complete?

f$)$ What is the probability that the project will be completed between the $15$ th and the $18$th week?

g$)$ What is the project completion date that yields a $95\%$ chance of completion?