Q$2,$ PERT $5$ marks$)$ As a new project that you need to manage, you were not able to get exact estimates for the activity durations. Instead, you were able to get three estimates for each activity: optimistic, most probable, and pessimistic as shown in the table below.

$\backslash$table$[[$Activity$,$Duration $($days$),$Predecessors,$\backslash$table$[[$Expected$],[$time$]],$Sigma,$($Sigma$)^2],[,$Optimistic,$\backslash$table$[[$Most$],[$probable$]],$Pessimistic,,,,$],[$A$,5,11,14,-,10\mathrm{.}5,1\mathrm{.}5,2\mathrm{.}25],[$B$,6,6,18,-,8,2,4],[$C$,1,10,13,-,9,2,4],[$D$,4,7,10,$A$,$ B$,7,1,1],[$E$,11,14,20,$B$,$ C$,14\mathrm{.}5,1\mathrm{.}5,2\mathrm{.}25],[$F$,7,16,16,$D$,$ E$,14\mathrm{.}5,1\mathrm{.}5,2\mathrm{.}25]]$

a$.$ For each activity, calculate its expected time and its standard deviation.

b$.$ Identify all paths in this project.

c$.$ For each path, calculate the path duration and its standard deviation.

d$.$ What is the probability that path A$,$ D$,$ F finishes in more than $29\mathrm{.}65$ days?

e$.$ Your manager is asking you for a time estimate for path A$,$ D$,$ F such that you are $95\mathrm{.}44\%$ sure that this path would finish in less than this time., PERT As a new project that you need to manage, you were not able to get exact estimates for the activity durations. Instead, you were able to get three estimates for each activity: optimistic, most probable, and pessimistic as shown in the table below