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A Probability Mass Function (PMF) is used to describe the distribution of a discrete random variable over its possible values. For our exercise, the demand for magazines each week is a discrete random variable, represented by \(X\). The PMF given helps us understand the likelihood (or probability) of each possible demand value occurring.

In this case, \(X\) can take values from 1 to 6, each with its own probability indicated in the table:

• Demand of 1 magazine: Probability = \(\frac{1}{15}\)
• Demand of 2 magazines: Probability = \(\frac{2}{15}\)
• Demand of 3 magazines: Probability = \(\frac{3}{15}\)
• Demand of 4 magazines: Probability = \(\frac{4}{15}\)
• Demand of 5 magazines: Probability = \(\frac{3}{15}\)
• Demand of 6 magazines: Probability = \(\frac{2}{15}\)

The sum of these probabilities equals 1, confirming that they are a complete distribution of potential outcomes. PMFs are essential in probability because they provide the precise probabilities needed to calculate expected values in decision making.

Statistical Decision Making involves analyzing probabilities and expected outcomes to make the best choice under uncertainty. In this exercise, the store owner uses statistical methods to decide how many magazines to order each week.

The choice to order either 3 or 4 magazines is crucial because it affects the net revenue, depending on what the actual demand \(X\) turns out to be. By considering each possible demand along with its probability, the owner can compute expected revenues for ordering different quantities:

• If all demanded magazines are sold, net revenue increases.
• For unsold magazines, there’s no gain, which underscores the need for careful decision making.

Ultimately, this decision-making process involves analyzing the expected revenue for each scenario and choosing the option that maximizes profitability based on probable outcomes. This smart decision-making strategy ensures the store owner makes choices aligned with potential profits.

Net Revenue Calculation is a critical component in determining profitability from selling magazines. Net revenue is defined as the difference between total sales revenue and the cost incurred for purchasing the magazines.

For this exercise:

• Each magazine costs the owner \\(2 and sells for \\)4.
• When 3 magazines are ordered: Different revenue calculations for each demand scenario range from subtracting costs of unsold magazines if demand is less or matching total sales if equal demand exists.
• When 4 magazines are ordered: Similar calculations apply, varying based on demand but ensuring maximum sales when demand matches or exceeds order quantity.

To find the Expected Revenue, we look at the possible revenues under different demand scenarios and weigh them according to their probabilities, which are derived from the PMF. The overall expectation helps choose the strategy (ordering 4 copies) that results in the highest average revenue over time. This calculation is crucial for making informed decisions that enhance business profitability.