Question $1(60$ Points$)$

Owner of an automobile dealership is concerned about the performance of his salesmen. From his past

experience with the dealership, he feels that the salesmen can be divided roughly into three categories,

which he calls "great", "good", and "poor". A "great" salesman sells the cars according to a Poisson process

at a rate of one every other day. A "good" salesman sells cars according to a Poisson process at a rate of

one every fourth day, and a "poor" salesman sells cars according to a Poisson process at a rate of one every

eighth day. The owner decides to express this information in terms of the uncertain quantity widetilde$(),$ which

represents the average sales per month $($each month contains $24$ working days$).$ Therefore, the three

possible values for tilde$()$ that he is considering are $12($the "great" salesman$),6($the "good" salesman$),$ and $3$

$($the "poor" salesman$).$ The owner of the dealership has just hired a new salesman. Based on past experience

with salesmen and on a personal interview with the new employee, the owner judges that the prior

distribution for tilde$()$ may be represented by following probabilities:

$P(tilde()=12)=0\mathrm{.}2,$

$P(widetilde()=6)=0\mathrm{.}5,$

$P(tilde()=3)=0\mathrm{.}3$

Suppose that the new salesman is offered a job for the next six months and he is

given a choice of three salary plans: $(1)$ a salary of $$5000$ plus a commission of $$90$ for each car

that he sells, $(2)$ a salary of $$8500$ with no commissions, or $(3)$ a commission of $$200$ for each

car that he sells, with no fixed salary. Construct the payoff and regret tables for this decision

problem.