$[30$ points$]$ Consider a project with following precedence relationships and activity times.

$\backslash$table$[[$Activity$,\backslash$table$[[$Immediate$],[$Predecessor$]],\backslash$table$[[$Optimistic$],[$Time$]],\backslash$table$[[$Most Likely$],[$Time$]],$Pessimistic Time$],[$A$,\frac{\text{:}2(4^{**}2\mathrm{.}5)6}{6}=3,2,2\mathrm{.}5,6],[$B$,10(4^{**}11)\frac{12}{6}=11,10,11,12],[$C$,5(4^{**}6\mathrm{.}5)\frac{11}{6}=7,5,6\mathrm{.}5,11],[$D$,11(4^{**}13)\frac{15}{6}=13,11,13,15],[$E$,8(4^{**}9\mathrm{.}5)\frac{14}{6}=10,8,9\mathrm{.}5,14],[$F$,4\mathrm{.}5(4^{**}5\mathrm{.}5)\frac{9\mathrm{.}5}{6}=6,4\mathrm{.}5,5\mathrm{.}5,9\mathrm{.}5],[$G$,3\mathrm{.}5(4^{**}5)\frac{6\mathrm{.}5}{6}=5,3\mathrm{.}5,5,6\mathrm{.}5],[$H$,6\mathrm{.}5(4^{**}7\mathrm{.}5)\frac{11\mathrm{.}5}{6}=8,6\mathrm{.}5,7\mathrm{.}5,11\mathrm{.}5]]$

a$.$ Calculate the expected activity duration $(t_e),$ variance for each activity, ES$,$ EF$,$ LS$,$ and LF times and the slack for each activity. Show the values in a table $($activity schedule$).$ What is the critical path for the project? Show all work to earn credit.

b$.$ Using MS Project $($or any other software$),$ create a Gantt chart for the project based on ES and EF times.

c$.$ What is the probability of completing the project in $40$ weeks? Show all work to earn credit.

d$.$ If the activity B can be finished in $10$ weeks, D in $11$ weeks, and $E$ in $8$ weeks, recalculate the expected project completion time. Show all work.