Question

e. [7.5 marks] The production rate of a company (D) could be 100 units per year with a probability of 0.4 or 125 units per year with a probability of 0.6. Moreover, the profit per unit sold (S) is either $5 with a probability of 0.35 or $7 with a probability of 0.65. What are the expected value and the standard deviation of the total annual profit (P)? i. 724.5 and 132 ii. 724.5 and 134.6 iii. 832.5 and 98.7 iv. 924.8 and 110.5

Answer 1

Ans. Option ii

The probability distribution table for total profit is,

Q = Quantity and Pq is its probability

g = profit per unit and Pg is its probability

G = total profit = Q*G and P = probability = Pq*Pg

Q. Pq. g. Pg. G. P

100 0.4 5 0.35 500 0.14

100 0.4 7 0.65 700 0.26

125 0.6 5 0.35 625 0.21

125. 0.6. 7. 0.65. 875. 0.39

E(G) = Expected value of total Gain = Sum of (P*G)

=> E(G) = 500*0.14 + 700*0.26 + 625*0.21 + 875*0.39

=> E(G) = 70 + 182 + 131.25 + 341.25 = $724.5

For standard deviation,

E(G^2) = Sum of (P*G^2) = 35000 + 127400 + 82031.25 + 298593.75

=> E(G^2) = 543025

Standard deviation = (E(G^2) - (E(G))^2)^0.5

=> Standard deviation = (543025 - 524900.25)^0.5

=> Standard deviation = 134.628

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