The orders received for grain by a farmer add up to X

Chapter , Problem 2

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The orders received for grain by a farmer add up to X tons, where X is a continuous random variable uniformly distributed over the interval (4, 7). Every ton of grain sold brings a profit of a, and every ton that is not sold is destroyed at a loss of a/3. How many tons of grain should the farmer produce to maximize his expected profit? Hint: Let Y (t) be the profit if the farmer produces t tons of grain. Then E 4 Y (t)5 = E 8 aX a 3 (t X)9 P (X < t) + E(at)P (X t).

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