$228$ Two modes of public transportation are available between two cities $150$ miles apart

A passenger could either ride the train or take a plane. The travel time for each transpor$-$

tation mode is randomly distributed according to the weather condition as follows:

During the winter months, the probability of bad weather on a given day of travel is $0\mathrm{.}2$

Suppose only travel time is considered in the choice of a specific mode of transportation.

Furthermore, because of limited seat capacitiey in both modes of transportation, a

passenger would have to decide on his travel plan at least one week ahead.

$($a$)$ Should train travel be pianned if the decision criterion is to minimize expected

travel time?

$($b$)$ Suppose the utilitylfunction for time is

$u(t)=-10t^2$

Which mode should be chosen if one wants to maximize expectod utility?

$($c$)$ What is the value of perfect information on the weather condition on the day of

travel, in terms of time units? Assume the same criterion as in $($b$).$