Thread: Expected value/ st dev/var

A merchant stocks a perishable item. He knows that on any given day he will have demand for two, three, or four of these items, with probabilities .2, .3, and .5. He buys items for $1 and sells them for $1.20. Any items left over at the end of the day represent a loss. How many items should the merchant stock to maximize his expected daily profit?

Work:
I think I need to use Tchebysheff's theorem but I'm unsure how to incorporate prices etc. into the problem.

There are only 3 scenarios, cant this be worked out manually?

R = Revenue
S = Stock level

Expected profit if you buy 2 item of stock
=E(R|S=2) - 2
=2.40 - 2
= 0.4

Expected Profit if you buy 3 items of stock
=E(R|S=3) - 3
=P(Demand for 3 or more)*3.6 + P(Demand for 2) * 2.4 -3
=0.8*3.6 + 0.2*2.4 - 3
=...

hmm don't we have to account for the fact that if the P(Demand for 2) then the 3rd item is a complete loss and we have to subtract another 1 from E(R|s=3)?

It is already accounted for. Profit is always Revenue - Cost.

The cost is fixed in each scenario, not random. In the "3 items of stock" scenario, the cos is 3. This is the last term in the expression i gave.