$3)$ GreenTech is managing a solar panel installation project for a commercial building. The project manager, Lisa Green, is concerned about delays because some tasks have unpredictable durations. The following table shows the optimistic time $($a$),$ most likely time $($m$),$ and pessimistic time $($b$)$ for each activity $($all in days$),$ along with their immediate predecessors: Assignment $1\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|$l$|$l$|\}\backslash$hline Activity & Optimistic Time $($a$)$ & Most Likely Time $($m$)$ & Pessimistic Time $($b$)$ & Immediate Predecessor$($s$)\backslash \backslash \backslash$hline A & $4$ & $5$ & $7$ & $-\backslash \backslash \backslash$hline B & $3$ & $4$ & $6$ & $-\backslash \backslash \backslash$hline C & $2$ & $3$ & $5$ & $-\backslash \backslash \backslash$hline D & $5$ & $7$ & $9$ & A $\backslash \backslash \backslash$hline E & $3$ & $5$ & $7$ & B$,$ C $\backslash \backslash \backslash$hline F & $6$ & $8$ & $10$ & D$,$ E $\backslash \backslash \backslash$hline G & $1$ & $2$ & $3$ & D $\backslash \backslash \backslash$hline H & $2$ & $4$ & $6$ & F $\backslash \backslash \backslash$hline I & $4$ & $6$ & $8$ & G $\backslash \backslash \backslash$hline J & $7$ & $9$ & $13$ & H$,$ I $\backslash \backslash \backslash$hline $\backslash$end$\{$tabular$\}$ a$)$ Calculate the expected time and variance for each activity using the PERT formulas: b$)$ Construct the project network and determine the expected project completion time and the critical path. c$)$ Compute the expected values for Earliest Start $($ES$),$ Earliest Finish $($EF$),$ Latest Start $($LS$),$ Latest Finish $($LF$),$ and slack for each activity. d$)$ Assuming activity durations are independent and normally distributed, calculate the probability that the project will finish in $40$ days or less.