Question $5[25$ Points$]$ Consider the following project:

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline $\backslash$multirow$\{2\}\{*\}\{$ Activity $\}$ & $\backslash$multirow$\{2\}\{*\}\{$ Depends on $\}$ & $\backslash$multicolumn$\{3\}\{|$c$|\}\{$ Activity Duration $($days$)\}\backslash \backslash$

$\backslash$cline $\{3-5\}$ & & Optimistic & Most Likely & Pessimistic $\backslash \backslash$

$\backslash$hline A & $-$ & $2$ & $5$ & $14\backslash \backslash$

$\backslash$hline B & $-$ & $3$ & $12$ & $21\backslash \backslash$

$\backslash$hline C & A & $5$ & $14$ & $17\backslash \backslash$

$\backslash$hline D & B & $2$ & $5$ & $8\backslash \backslash$

$\backslash$hline E & C$,$D & $1$ & $4$ & $7\backslash \backslash$

$\backslash$hline F & B & $6$ & $15$ & $30\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

a$)$ Determine expected duration $\backslash (\backslash$mathrm$\{$t$\}\_\{\backslash$mathrm$\{$e$\}\}\backslash )$ and $\backslash (\backslash$sigma$\_\{\backslash$mathrm$\{$Te$\}\}\backslash )$ for each activity. $(5$ points$)$

b$)$ Determine $\backslash ($ T$\_\{$e$\}\backslash )$ and $\backslash (\backslash$sigma$\_\{$T$\_\{$e$\}\}\backslash )(5$ points$)$

c$)$ What is the probability that: $(15$ points$)$

i$.$ The project will be completed in $25$ days.

ii$.$ The project will be completed in more than $30$ days.

iii. The project will take between $26$ and $30$ days.

d$)$ If the owner wishes to work on a probability of completion of $0\mathrm{.}8,$ what will be the earliest project time? $(5$ points$)$