Suppose the demand for newspapers is equally likely to be any integer between 0 and 24, i. e., x ~ unif{0,1,2,…,24} and we buy them in the morning for $.30, we sell them for $1, and we get $.10 for any newspapers leftover at the end of the day from the recycling company. how many newspapers should we obtain to sell during the day?

We should obtain 19 newspapers to maximize the expected profits.

Explanation:

In this problem we have to optimize the quantity of newspaper we buy to maximize the profit.

We can divide the analysis in two parts:

1) If the amount of newspaper sold (S) is more than the newspaper we bougth (B), the profit is .

This occurs with probability , as S is uniformly distributed.

2) If the amount of newspaper sold (S) is less than the newspaper we bougth (B) we have a profit of:

This happens with a probability of .

As we have S in our profit equation, we have to calculate the expected value of S to estimate the profit in function of B.

If S is smaller than B, then it is uniformly distributed between 0 and B. The expected value can be calculated as:

Then, the expected profit formula for S<B is

The total profit tcan be calculated by adding the two parts analized previously: when S>B and when S<B, affected by the probability of occurrence:

To maximize the profit G, we derive and equal zero

We should obtain 19 newspapers to maximize the expected profits.