Respuesta :
Answer:
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 [tex]G_1=(1.0-0.3)*B=0.7*B[/tex].
This occurs with probability [tex]P(S>B)=1-B/24[/tex], 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:
[tex]G_2=1.0*S+0.1*(B-S)-0.3*B=0.9S-0.2B[/tex]
This happens with a probability of [tex]P(S≤B)=B/24[/tex].
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:
[tex]E[S]=\frac{B-0}{2} =B/2[/tex]
Then, the expected profit formula for S<B is [tex]G_2=0.9S-0.2B=0.45B-0.2B=0.25B[/tex]
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:
[tex]G=P(S>B)*G_1+P(S\leq B)*G_2\\\\G=(1-\frac{B}{24})*(0.7B)+(\frac{B}{24} )*0.25B\\\\G=0.7B-\frac{0.7}{24}B^2+\frac{0.25}{24}B^2\\\\G=-0.01875B^2+0.7B[/tex]
To maximize the profit G, we derive and equal zero
[tex][/tex]
[tex]\frac{dG}{dB}= -0.01875*2*B+0.7=0\\\\-0.0375B+0.7=0\\\\B=0.7/0.0375=18.66 \approx 19[/tex]
We should obtain 19 newspapers to maximize the expected profits.
