$[20$ points$]$ Environment Recycling, Inc. must clean up a large automobile tire dump under a state environmental cleanup contract. The project activity times are not certain due to various reasons and the following table shows the time estimates for activity times. Activity times are assumed to follow a Beta distribution:

$\backslash$table$[[$Activity$,\backslash$table$[[$Immedlate$],[$Predecessor$]],\backslash$table$[[$Optimistic Time$],[($weeks$)]],\backslash$table$[[$Most Probable$],[$Time $($weeks$)]],\backslash$table$[[$Pessimistic Time$],[($weeks$)]]],[$A$,$none,$6,8,10],[$B$,$none,$8,10,12],[$C$,$A$,$ B$,4,6,8],[$D$,$B$,4,5,6],[$E$,$C$,10,12,14],[$F$,$C$,6,7,8],[$G$,$D$,$ E$,8,12,16],[$H$,$F$,$ G$,6,8,10]]$

MGMT $3106$: Management Science and Operations Management

a$.$ Calculate the expected time and variance for each activity.

b$.$ Create the activity schedule and calculate ES$,$ EF$,$ LS$,$ and LF times. Activity schedule is a table with ES$,$ EF$,$ LS$,$ LF times for each activity.

c$.$ Calculate the slack for each activity.

d$.$ Determine the estimated project completion time and the standard deviation of the critical path. Show all calculations.

e$.$ Explain why this project has an estimated project completion time and a standard deviation.