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A consumer electronics store stocks four alarm clock radios. If it has fewer than four clock radios available at the end of a week, the store restocks the item to bring the in-stock level up to four. If weekly demand is greater than the four units in stock, the store loses sales. The radio sells for $25 and costs the store $15.

I calculated the expected value of expected weekly demand for the alarm clock radio is 3.6.

For the question "What is the expected weekly profit from the sale of the alarm clock radio? (Remember: There are only four clock radios available in any week to meet demand.)"

Do I calculate the sales and cost using the expected value or using all four alarm clock radio?

Second question "On average, how much profit is lost each week because the radio is not available when demanded?"

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  • $\begingroup$ Missing information: What is the distribution of the demand? (The expected demand is not enough.) $\endgroup$ Commented May 2, 2017 at 5:30

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Let $D$ be the demand, and $P$ the profit.

You profit $\$10$ on each clock sold, but cannot sell more than 4, so clearly:

$$P = \begin{cases}10 D &:& D\in\{0,1,2,3,4\}\\ 40 &:& D\in\{5,...\}\\ 0 &:& \text{else}\end{cases}$$

Let $T$ be the theoretical profit if you were able to instantly stock clocks on demand.

$$T = \begin{cases}10 D &:& D\in\{0,...\}\\ 0 &:& \text{else}\end{cases}$$

Obviously the difference is: $$T-P = \begin{cases}10(D-4)&:& D\in\{5,...\}\\ 0 & \text{else}\end{cases}$$

You need to know the distribution of the demand to calculate the expectations.

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