Large Industrial Firm Purchases Several New Word Processors

Large Industrial Firm Purchases Several New Word Processors At The

A large industrial firm purchases several new word processors at the end of each year. The number of processors purchased depends on the previous year's repairs, with the probability distribution: P(X=1) = 1/10, P(X=2) = 2/5. The cost per processor is $1200, with a $50 credit for each processor purchased, plus a fixed cost of $600 regardless of quantity. The task is to determine the expected spending on new word processors for the firm.

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To calculate the expected expenditure of the firm on new word processors, we need to analyze the probability distribution of the number of processors purchased, the costs associated, and the credits provided. The purchase count, represented by the random variable X, has the following probabilities: P(X=1) = 1/10 and P(X=2) = 2/5. Although the distribution implies only these two points, typically, it should be expanded to include all possible values, but based on the provided data, we analyze these two outcomes.

The cost for each processor is $1200. However, a discount of $50 per processor is credited, which effectively reduces the cost per processor to $1150 ($1200 - $50). A fixed cost of $600 is incurred regardless of the number of processors purchased. The total cost for purchasing X processors is thus:

  • Total cost = Fixed cost + (Cost per processor - Credit) × Quantity
  • Total cost = $600 + ($1200 - $50) × X = $600 + $1150 × X

To find the expected expenditure, we calculate the expected value of the total cost as:

Expected Cost = E[Total cost] = Σ P(X = x) × Total cost when X = x

Calculating for each possible value of X:

  • X = 1: Probability = 1/10
  • X = 2: Probability = 2/5 = 4/10

Expected cost:

= (1/10) × [$600 + $1150 × 1] + (4/10) × [$600 + $1150 × 2]

= (1/10) × [$600 + $1150] + (4/10) × [$600 + $2300]

= (1/10) × [$1750] + (4/10) × [$2900]

= $175 + $1160

= $1335

Therefore, the firm is expected to spend approximately $1,335 annually on new word processors based on the given probability distribution and costs.

References

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