2.2.2 Examples of Linear Combinations of Random Variables

Example 1: Total Cost of Production

• Situation: Calculating the total cost of a product based on material and labor costs, both of which are random variables.
• Random Variables:
• X1: Units of material used (random)
• X2: Units of labor used (random)

• Linear Combination:

• a1​, a2: Per-unit costs of material and labor, respectively.

• Result: Y represents the total cost, influenced by the random quantities of material and labor.

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Example 2: Total Monthly Profit of a Retail Chain

• Situation: A retail chain operates several stores; monthly sales at each are random.

• Random Variables:

  Monthly sales in units at each of n stores.

• Linear Combination:

• 

  Profits per unit sold in each store.

• Result: Y represents the total monthly profit across all stores, combining individual profits adjusted by sales volumes.

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Example 3: Total Monthly Expenses of a Business

• Situation: A company incurs monthly expenses in various categories like personnel, marketing, etc., each being a random variable due to fluctuating costs.

• Random Variables:

  Expenditures in each category.

• Linear Combination:

• Result: Y represents the total monthly expense of the business, aggregating all categorical expenses.

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Example 4: Average Sales Commission

• Situation: Sales commissions are collected by different sales reps, each commission being a random variable.

• Random Variables:

  Commissions collected by n sales reps.

• Linear Combination:

• Result:
  Y is the average sales commission per sales rep for that month.

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