Problem $3$

The product development group of a tech company has been working on a new computer software program. The activities, their immediate predecessors, and time estimates $($in weeks$)$ for the software development project are given in the following table:

$\backslash$table$[[$Activity$,\backslash$table$[[$Immediate$],[$Predecessors$]],$Time $($weeks$)],[$Optimistic$,$Most Likely,Pessimistic$],[$A$,-,3,4,5],[$B$,$A$,3,3\mathrm{.}5,7],[C,$B$,4,5,6],[$D$,C,2,3,4],[$E$,$B$,5,9,13],[F,$B$,7\mathrm{.}5,8\mathrm{.}5,12\mathrm{.}5],[G^{},E,4\mathrm{.}5,6,7\mathrm{.}5],[H,E,5,6,13],[$I$,D,G,2,2\mathrm{.}5,6],[J,F,H,$I,$4,5,6]]$

a$.$ Calculate the expected durations and the variance $($or standard deviation$)$ for each activity.

b$.$ Applying PERT, find the expected project completion time and the project standard deviation?

c$.$ What is the probability $($likelihood$)$ that the project will be completed $($so that the company may introduce the new product$)$ within $32$ weeks?

d$.$ If you wanted to be $99\%$ confident of completing the project on time, what would be its completion time target?PLEASE SHOW ALL WORK!!