At the beginning of every month, a pharmacist orders an amount of a certain costly medicine that comes in strips of individually packed tablets. The wholesale price per strip is 100, and the retail price per strip is 400. The medicine has a limited shelf life. Strips not purchased by month’s end will have reached their expiration date and must be discarded. When it so happens that demand for the item exceeds the pharmacist’s supply, he may place an emergency order for 350 per strip. The monthly demand for this medicine takes on the possible values 3, 4, 5, 6, 7, 8, 9, and 10 with respective probabilities 0.3, 0.1, 0.2, 0.2, 0.05, 0.05, 0.05, and 0.05. The pharmacist decides to order eight strips at the start of each month. What are the expected value and the standard deviation of the net profit made by the pharmacist on this medicine in any given month?
Any hints on where to start?
Since this is a discrete distribution, just consider all possible cases:
For $demand < 9$, his profit is $400*demand - 800$.
For $demand = 9, 10$, his profit is $2400 + 50*(demand - 8)$.
So, you can write these values out for each of the values of $demand$. From here you should be able to figure out the expected value and variance.
Let me know if you have any questions about how I got the profit equations or how to calculate mean or variance.
Note that the implicit assumption here is that if the pharmacist has a demand of 9 or 10, he is required to fulfill it by ordering emergency strips.
Irrelevant comment, but two things I've learned from this problem are that, one, pharamacists are extremely rich (he's making a profit even if the demand is minimal), and two, he might be able to increase his expected profit by ordering a strip or two less, just based on the demand probabilities. Might be an interesting thing you could look into.