2. 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[Y (t)] = E[aX (t - X)] P(X < t) + E(at)P(X t).