A small shop orders copies of a certain magazine each week L

A small shop orders copies of a certain magazine each week. Let X=demand for the magazine with pmf

The shop owner pays $1.50 for each copy of the magazine and the price to customers is $2.50. If the magazines left at the end of the week have no salvage value, is it better to order one, two, three or four copies of the magazine?

Hint: Introduce random variables;Y_k=the number of magazines sold if k magazines are ordered, R_k=the net profit if k magazines are ordered, k=1,2,3,4 and find their probability distributions and expected value.

Please help and explain how you arrived at your answer, Im not sure where to start

Solution

Probability Distribution :

P(X>=1) = 0.2+0.3+0.4+0.1 =1.0

P(X>=2) = 0.3+0.4+0.1 =0.8

P(X>=3) = 0.4+0.1 =0.5

P(X>=4) = 0.1

Now,

>>

Cost of 1 magazine = 1*1.5 =1.5

Expected return on 1 magazines = 1 * 2.5 * 1.0 = 2.5

Profit on 1 = 2.5 -1.5 = $1.0

>>

Cost of 2 magazine = 2*1.5 = 3

Expected return on 2 magazines = 2 * 2.5 * 0.8 = 4

Profit on 2 = 4 -3 = $1.0

>>

Cost of 3 magazine = 3*1.5 = 4.5

Expected return on 3 magazines = 3 * 2.5 * 0.5 = 3.75

Loss on 3 = 4.5 -3.75 = $0.75

>>

Cost of 4 magazine = 4*1.5 = 6.0

Expected return on 4 magazines = 4 * 2.5 * 0.1 = 1

Loss on 4 = 6.0 -1 = $5.0