Problem $2.$ Best Price in Town $($BPT$)$ is an electronics store in Ann Arbor. They are

currently deciding how many units of a new digital camera, called Model Z$,$ to order. Let

$Q$ denote this stock level choice. BPT has a single ordering opportunity for Model Z$.$ Once

Model $Z$ arrives at BPT$,$ it will be sold for three months, and then will be replaced by some

future model. At the end of the three months, the leftover inventory for Model $Z$ will have

no value, and will be discarded.

The total customer demand for Model $Z$ over the next three months in Ann Arbor is a

random variable, denoted by $D.$ Let $F$ denote the cumulative distribution function of $D,$

and $f$ the probability density function. Of all the customers in Ann Arbor who want to buy

Model $Z,$ only a fraction $k$ will demand the product from BPT$.$ The rest will go to other

electronic stores. $($For example, if $D$ turns out to be $100$ and $k=0\mathrm{.}6,$ then the demand for

Model $Z$ at BPT will be $60.)$

The unit retail price of the camera is $p,$ and the unit purchase price that BPT pays to the

supplier of the camera is $c.$

$($a$)$ Write an expression for the expected sales of Model $Z$ at BPT for a given stock level

choice Q$,$ i$.$e$.$ the expected number of Model $Z$ that BPT will sell in the next three months.

$($This expression should be detailed enough that if we knew BPT$\text{'}$s stock level choice $($Q$),$

the fraction of total demand that BPT gets $(k),$ the functions $F$ and $f,$ and had a tool at our

disposal to evaluate integrals, then we could evaluate this expression.$)$

$($b$)$ Using your answer to part $($a$),$ write an expression for the expected profit BPT will

make from Model Z$.$

$($c$)$ Using your answer to part $($b$),$ derive a relationship that needs to be satisfied by

BPT$\text{'}$s optimal stock level.