$($c$)$ The airline considers a penalty function in deciding how many passengers to overbook.

If more than $50$ passengers show up$,$ a penalty of $$300$ per extra passenger is considered

$($loss of customer goodwill, bumping passengers to a different flight etc.$).$ If less than $50$

passengers show up$,$ a penalty of $$250$ per empty seat is considered $($opportunity cost

etc.$).$ What is the expected value of the penalty function? $(20$ points$)$