Problem $9[8$ pts$]$

Assume you are staying late today for tomorrow's exam. The exam is at $9$ AM$.$ You know the distribution of time from getting up to arriving at school. $($since traffic time might be random$).$ Because you are very sleepy, the performance of your exam depends on how much you can sleep, so you can get $0\mathrm{.}2$ more point every $1$ more minute you sleep. If you arrive at the exam late, because of the limited time, every $1$ more minute you are late for, on average you will lose $0\mathrm{.}8$ point. What's the optimal time for you to get up in the morning?

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

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Time to $\backslash \backslash$

take from $\backslash \backslash$

getting up $\backslash \backslash$

to school

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

$15\backslash \backslash$

min

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

$16\backslash \backslash$

min

$\backslash$end$\{$tabular$\}$ & $17$ min & $\backslash$begin$\{$tabular$\}\{$l$\}$

$18\backslash \backslash$

min

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

$19\backslash \backslash$

min

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

$20\backslash \backslash$

$\backslash (\backslash$min $\backslash )$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

$21\backslash \backslash$

$\backslash (\backslash$min $\backslash )$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

$22\backslash \backslash$

$\backslash (\backslash$min $\backslash )$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

$23\backslash \backslash$

$\backslash (\backslash$min $\backslash )$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

$24\backslash \backslash$

min

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

$\backslash$hline probability & $0\mathrm{.}1$ & $0\mathrm{.}1$ & $0\mathrm{.}1$ & $0\mathrm{.}1$ & $0\mathrm{.}1$ & $0\mathrm{.}1$ & $0\mathrm{.}1$ & $0\mathrm{.}1$ & $0\mathrm{.}1$ & $0\mathrm{.}1\backslash \backslash$

$\backslash$hline

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