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](https://tex.z-dn.net/?f=%20E%28R%29%20%3D%20%5Cfrac%7B1%7D%7B15%7D%20%2A%28-2%29%20%2B%5Cfrac%7B2%7D%7B15%7D%20%2A%282%29%20%2B%5Cfrac%7B3%7D%7B15%7D%2A%286%29%20%2B%20%5Cfrac%7B4%7D%7B15%7D%2A%286%29%20%2B%5Cfrac%7B3%7D%7B15%7D%2A%286%29%20%2B%5Cfrac%7B2%7D%7B15%7D%2A%286%29%20%3D%20%5Cfrac%7B74%7D%7B15%7D%3D4.93)
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](https://tex.z-dn.net/?f=%20E%28R%29%20%3D%20%5Cfrac%7B1%7D%7B15%7D%20%2A%28-4%29%20%2B%5Cfrac%7B2%7D%7B15%7D%20%2A%280%29%20%2B%5Cfrac%7B3%7D%7B15%7D%2A%284%29%20%2B%20%5Cfrac%7B4%7D%7B15%7D%2A%288%29%20%2B%5Cfrac%7B3%7D%7B15%7D%2A%288%29%20%2B%5Cfrac%7B2%7D%7B15%7D%2A%288%29%20%3D%20%5Cfrac%7B80%7D%7B15%7D%3D5.33)
As as we can see we have a higher expected value for the case with 4 magazines.