Question

A small market orders copies of a certain magazine for its magazine rack each week. Let X 5 demand for the magazine, with pmf Suppose the store owner actually pays $2.00 for each copy of the magazine and the price to customers is $4.00. If magazines left at the end of the week have no salvage value, is it better to order three or four copies of the magazine? [Hint: For both three and four copies ordered, express net revenue as a function of demand X, and then compute the expected revenue.]

Answer 1

Answer:

See explanation below.

Explanation:

Let X the random variable that represent the demand for the magazine, the pmf for X is given by:

X       1            2           3          4        5        6      

P(X)  1/15      2/15       3/15     4/15   3/15     2/15

3 magazines

For this case the total spent is 2*3 = $ 6

And the net revenue for this case would be:

$4-$6 = -$2 , X=1 (demand 1)

$4*2-$6 = $2 , X=2 (demand 2)

$4*3-$6 = $6 , X=3 (demand 3)

For the values of X=4,5,6 the net revenue will be $6 since the number of magazines is 3

And the expected value for the net revenue would be:

$E(R) = \frac{1}{15} *(-2) +\frac{2}{15} *(2) +\frac{3}{15}*(6) + \frac{4}{15}*(6) +\frac{3}{15}*(6) +\frac{2}{15}*(6) = \frac{74}{15}=4.93$

4 magazines

For this case the total spent is 2*4 = $ 8

And the net revenue for this case would be:

$4-$8 = -$4 , X=1 (demand 1)

$4*2-$8 = $0 , X=2 (demand 2)

$4*3-$8 = $4 , X=3 (demand 3)

$4*4-$8 = $8 , X=4 (demand 4)

For the values of X=5,6 the net revenue will be $8 since the number of magazines is 4

And the expected value for the net revenue would be:

$E(R) = \frac{1}{15} *(-4) +\frac{2}{15} *(0) +\frac{3}{15}*(4) + \frac{4}{15}*(8) +\frac{3}{15}*(8) +\frac{2}{15}*(8) = \frac{80}{15}=5.33$

As as we can see we have a higher expected value for the case with 4 magazines.