Problems of choice with immediate effect under conditions of uncertainty

Question

A farm, in order to commercialize a product, may select between two intermediaries, which offer the following conditions:

A) A fixed cost of 2000 dollars for any level of production; B) A variable cost equal to 10% of revenue.

Each kilogram of product is sold at $ 1.

Determine which broker choose if "production to sell" in a certain year is a random variable with the following probability distribution:

Quantity in tons: 10; Probability 10% 
Quantity in tons: 20; Probability 25% 
Quantity in tons: 30; Probability 40% 
Quantity in tons: 40; Probability 15% 
Quantity in tons: 50; Probability 10%

Which of the two alternatives is more risky for the farm?

The reference solution of my book is the following: It is necessary to calculate the average earnings: M (G(A)) = 28000, M (G(B)) = 27000; it is more convenient the alternative A which, however, is more risky. The risk is to pay a cost too high in case the production to market is low. Risk is measured by standard deviation sigma (G (A)) = 13038, sigma (G (B)) = 11735

And this is my partial solution:

I calculate M(G(A)) which is the MEAN of the Gain for alternative "A":

TONS;    REVENUE;         COST;        GAIN
10;   10000 DOLLARS; 2000 DOLLARS; 8000 DOLLARS;
20;   20000 DOLLARS; 2000 DOLLARS; 18000 DOLLARS;
30;   30000 DOLLARS; 2000 DOLLARS; 28000 DOLLARS;
40;   40000 DOLLARS; 2000 DOLLARS; 38000 DOLLARS;
50;   50000 DOLLARS; 2000 DOLLARS; 48000 DOLLARS.

By multiplying every gain for the respective probability, I obtain:

8000 * 0.10 + 18000 * 0.25 + 28000 * 0.40 + 38000 * 0.15 + 48000 * 0.10 = M(G(A) = 27000

WHY my book says 28000??? First result that makes me crazy... :-(

Could you help me for this?

Thank you for considering my request.

Observation: You've used the right formula, and using it with those numbers gives that answer. Conclusion: Your book is wrong. — Graham Kemp, Oct 12 '14 at 09:03
And what about sigma(G(A)) and sigma(G(B))? My result is wrong again, could you explain it to me please? — Ottavio Spadaccini, Oct 12 '14 at 09:18

score 1 · Accepted Answer · answered Oct 12 '14 at 11:58

Using those numbers you should have gotten:

$$\boxed{\begin{array}{|r|r|} \hline Q & \mathsf P(Q) & A: Q-2\,000 & B: 0.9\, Q & A\,\mathsf P(Q) & A^2\,\mathsf P(Q) & B\,\mathsf P(Q) & B^2\,\mathsf P(Q) \\ \hline 10\,000 & 10\% & 8\,000 & 9\,000 & 800 & 6\,400\,000 & 900 & 8\,100\,000 \\ \hline 20\,000 & 25\% & 18\,000 & 18\,000 & 4\,500 & 81\,000\,000 & 4\,500 & 81\,000\,000 \\ \hline 30\,000 & 40\% & 28\,000 & 27\,000 & 11\,200 & 313\,600\,000 & 10\,800 & 291\,600\,000 \\ \hline 40\,000 & 15\% & 38\,000 & 36\,000 & 5\,700 & 216\,600\,000 & 5\,400 & 194\,400\,000 \\ \hline 50\,000 & 10\% & 48\,000 & 45\,000 & 4\,800 & 230\,400\,000 & 4\,500 & 202\,500\,000 \\ \hline \hline & & & \sum & 27\,000 & 848\,000\,000 & 26\,400 & 777\,600\,000 \\ \hline \end{array}} \\[2ex] \begin{align} \overline A &= 27\,000 \\ \sigma_{\small A}^2 &= 848\,000\,000-27\,000^2 \\ &=119\,000\,000 \\ \sigma_{\small A}&\approx 10\,908.{\small 7\ldots} \\[2ex] \overline B &= 26\,400 \\ \sigma_{\small B}^2 &= 777\,600\,000-26\,400^2 \\ &=80\,640\,000 \\ \sigma_{\small B} &\approx 8\,980.{\small 0\ldots} \end{align}$$

Thank you very much for your kind help!! Answer ACCEPTED, thanks again friend! ;-) — Ottavio Spadaccini, Oct 13 '14 at 04:27

Problems of choice with immediate effect under conditions of uncertainty

1 Answers1