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A grocery store has n watermelons to sell and makes $1.00 on each sale. Say the number
Chapter 3, Problem 3.1-6(choose chapter or problem)
A grocery store has n watermelons to sell and makes $1.00 on each sale. Say the number of consumers of these watermelons is a random variable with a distribution that can be approximated by
\(f(x)=\frac{1}{200}, \quad 0<x<200\)
a pdf of the continuous type. If the grocer does not have enough watermelons to sell to all consumers, she figures that she loses $5.00 in goodwill from each unhappy customer. But if she has surplus watermelons, she loses 50 cents on each extra watermelon. What should n be to maximize profit? Hint: If \(X \leq n\), then her profit is \((1.00) X+(-0.50)(n-X)\); but if X > n, her profit is (1.00)n+(-5.00)(X-n). Find the expected value of profit as a function of n, and then select n to maximize that function.
Questions & Answers
QUESTION:
A grocery store has n watermelons to sell and makes $1.00 on each sale. Say the number of consumers of these watermelons is a random variable with a distribution that can be approximated by
\(f(x)=\frac{1}{200}, \quad 0<x<200\)
a pdf of the continuous type. If the grocer does not have enough watermelons to sell to all consumers, she figures that she loses $5.00 in goodwill from each unhappy customer. But if she has surplus watermelons, she loses 50 cents on each extra watermelon. What should n be to maximize profit? Hint: If \(X \leq n\), then her profit is \((1.00) X+(-0.50)(n-X)\); but if X > n, her profit is (1.00)n+(-5.00)(X-n). Find the expected value of profit as a function of n, and then select n to maximize that function.
ANSWER:Step 1 of 3
Given n represents the number of watermelons available for sales,
\(f(x)=\frac{1}{200}, 0<x<200\)
Let X is the number of customers that want to buy watermelons and let P represents the profit.
Let’s assume that \(P=X-0.50(n-X)\)
For profit of 1 on each sale and lose 5 for each customer then, \(P=n-5(X-n)\)