Chapter 3.1, Problem 6E

A grocery store can order n watermelons from a supplier. The grocer will make $1.00 on each sale of a watermelon, and she will limit sales to one per customer. Say X, the number of customers who wish to buy a watermelon, is a random variable with a distribution that can be approximated by $f\left(x\right)=\frac{1}{200},0<x<200$, a pdf of the continuous type. If the grocer does not have enough watermelons to sell to all customers who wish to buy one, she figures that she will lose $5.00 in goodwill from each unhappy customer. But if she doesn’t sell all the watermelons she orders, she loses 50 cents on each one not sold. What should ii be to maximize the grocer’s expected profit? HINT. If $X\le n$, then her profit is $\left(1.00\right)X+\left(-0.50\right)\left(n-X\right)$; but if $X>n$, her profit is $\left(1.00\right)X+\left(-0.50\right)\left(X-n\right)$.

Find the expected value of profit as a function of n, and then select n to maximize that function.