91Ó°ÊÓ

Suppose that the number of plants of a particular type found in a rectangular sampling region (called a quadrat by ecologists) in a certain geographic area is an rv \(X\) with pmf $$ p(x)= \begin{cases}\mathrm{c} / x^{3} & x=1,2,3, \ldots \\ 0 & \text { otherwise }\end{cases} $$ Is \(E(X)\) finite? Justify your answer (this is another distribution that statisticians would call heavy-tailed). 35\. A small market orders copies of a certain magazine for its magazine rack each week. Let \(X=\) demand for the magazine, with pmf \begin{tabular}{l|llllll} \(x\) & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline\(p(x)\) & \(\frac{1}{15}\) & \(\frac{2}{15}\) & \(\frac{3}{15}\) & \(\frac{4}{15}\) & \(\frac{3}{15}\) & \(\frac{2}{15}\) \end{tabular} Suppose the store owner actually pays \(\$ 2.00\) for each copy of the magazine and the price to customers is \(\$ 4.00\). If magazines left at the end of the week have no salvage value, is it better to order three or four copies of the magazine? [Hint: For both three and four copies ordered, express net revenue as a function of demand \(X\), and then compute the expected revenue.]

Step by step solution

01

Calculate Expected Revenue for Ordering Three Copies

The net revenue from selling magazines can be calculated by considering how many are sold for various demands when ordering three copies. Since the store pays \\(2.00 per copy and sells at \\)4.00, the profit per sold copy is \\(2.00.- If demand \(X = 1\): Revenue is \\)2.00 per copy, total \(2 \times 1 = \\(2.00\).- If demand \(X = 2\): Revenue is \\)2.00 per copy, total \(2 \times 2 = \\(4.00\).- If demand \(X = 3\) or more: All three copies are sold, total \(2 \times 3 = \\)6.00\).Calculate expected revenue as:\[ E(R_3) = \sum_{x=1}^{6} R(x) \cdot p(x) \]Where \(R(x)\) represents the revenue based on different demands.

02

Calculate Revenue Values for Ordering Three Copies

- Revenue for \(X = 1\): \\(2.00 with \(p(x) = \frac{1}{15}\).- Revenue for \(X = 2\): \\)4.00 with \(p(x) = \frac{2}{15}\).- Revenue for \(X \geq 3\): \$6.00 with cumulative probability \((3/15 + 4/15 + 3/15 + 2/15) = \frac{12}{15}\).Substitute these into the expected revenue equation:\[ E(R_3) = 2 \times \frac{1}{15} + 4 \times \frac{2}{15} + 6 \times \frac{12}{15} \]

03

Compute Expected Revenue for Three Copies

Calculate \(E(R_3)\) using the formula:\[ E(R_3) = \frac{2}{15} + \frac{8}{15} + \frac{72}{15} = \frac{82}{15} \approx 5.47 \]So, the expected revenue for ordering three copies is approximately \$5.47.

04

Calculate Expected Revenue for Ordering Four Copies

The net revenue is calculated similarly for ordering four copies:- If demand \(X = 1\): Revenue is \\(2.00, total \(2 \times 1 = \\)2.00\).- If demand \(X = 2\): Revenue is \\(4.00, total \(2 \times 2 = \\)4.00\).- If demand \(X = 3\): Revenue is \\(6.00, total \(2 \times 3 = \\)6.00\).- If demand \(X = 4\) or more: All four copies sold, total \(2 \times 4 = \$8.00\).Calculate expected revenue as: \[ E(R_4) = \sum_{x=1}^{6} R(x) \cdot p(x) \]

05

Calculate Revenue Values for Ordering Four Copies

- Revenue for \(X = 1\): \\(2.00 with \(p(1) = \frac{1}{15}\).- Revenue for \(X = 2\): \\)4.00 with \(p(2) = \frac{2}{15}\).- Revenue for \(X = 3\): \\(6.00 with \(p(3) = \frac{3}{15}\).- Revenue for \(X \geq 4\): \\)8.00 with cumulative probability \((4/15 + 3/15 + 2/15) = \frac{9}{15}\).Substitute these into the expected revenue equation:\[ E(R_4) = 2 \times \frac{1}{15} + 4 \times \frac{2}{15} + 6 \times \frac{3}{15} + 8 \times \frac{9}{15} \]

06

Compute Expected Revenue for Four Copies

Calculate \(E(R_4)\) using the formula:\[ E(R_4) = \frac{2}{15} + \frac{8}{15} + \frac{18}{15} + \frac{72}{15} \]\[ E(R_4) = \frac{100}{15} \approx 6.67 \]So, the expected revenue for ordering four copies is approximately \$6.67.

07

Conclusion: Determine Optimal Order Quantity

Compare the expected revenues:- Three copies: \\(5.47- Four copies: \\)6.67Ordering four copies gives a higher expected revenue. Therefore, it is better to order four copies.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value Calculation

Expected value is a key concept in probability theory, crucial for decision making under uncertainty. To understand expected value, consider it as the weighted average of all possible outcomes of a random variable. In simpler terms, it's what you can "expect" to happen on average if you repeat an experiment many times.

For our magazine example, the expected revenue when ordering a specific number of copies is calculated by multiplying each possible revenue outcome by its probability, and summing up all those products. This method allows us to predict which ordering strategy yields the highest average revenue, guiding the store owner towards an optimal decision.

It's important to note that expected value provides a long-term average and does not guarantee a specific outcome on a single trial. Nonetheless, it's a powerful tool for assessing different strategies and their potential financial benefits.

Demand Forecasting

Demand forecasting is a critical element in supply chain management and business strategy. It involves predicting the future demand for products or services to optimize supply decisions. In the context of our exercise, the store owner uses demand forecasting to determine how many magazine copies to purchase weekly.

Demand forecasting can be approached through historical data, probability distributions, and understanding market trends. For the magazines, the probability distribution (pmf) represents historical demand data, helping estimate future sales. This data-driven approach aids in determining the most cost-effective stock levels to maximize revenue while minimizing wasted resources.

Effective demand forecasting balances stocking too little (leading to missed sales) or too much (resulting in excess unsold stock). Therefore, understanding demand probabilities is key to making optimal purchasing decisions.

Decision Making Under Uncertainty

Decision making under uncertainty is an inevitable aspect of running a business, particularly concerning inventory and stock decisions. In the magazine example, the owner faces uncertainty in predicting exact demand each week, but must still decide on the number of copies to order.

When making decisions under uncertainty, one typically weighs the risks and rewards of multiple scenarios. Here, the owner's goal is to maximize expected revenue while considering the probability of various demand levels. Using expected value calculations assists in choosing an option that yields the best estimated profit.

A key takeaway is that while uncertainty can never be fully eliminated, probabilistic models provide a foundation for informed decision making. By analyzing potential outcomes and their probabilities, businesses can make strategic choices, minimizing risks and harnessing opportunities to improve overall performance.