$5\mathrm{.}5$ In the following table, task durations are given in weeks. The estimates were made at the $95$ percent level $($see Section $5\mathrm{.}2,$ Calculating Probabilistic Activity Times subsection$).$

$\backslash$table$[[$Activity$,$Predec.,Opt.,Normal,Pess.$],[$a$,-,2,4,6],[$b$,-,3,5,9],[$c$,$A$,4,5,7],[$d$,$A$,4,6,10],[$e$,$b$,$ c$,4,5,7],[$f$,$D$,3,4,8],[$g$,$e$,3,5,8]]$

a$)$ Find the expected time and variance for each task.

b$)$ Draw the network $($either AOA or AON$)$ and find the path with the longest expected time. What is the expected time to complete this path?

c$)$ Find the probability using the analytical approach that the path with the longest expected time will be completed in $23$ weeks.

d$)$ Find the probability that the other paths will be completed in $23$ weeks.

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