6. Normal Distribution & Continuous Random Variables

Uniform Distribution

6. Normal Distribution & Continuous Random Variables

Uniform Distribution

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Multiple Choice

A regional sales manager is forecasting sales for a new energy-efficient refrigerator. Based on market research she estimates that weekly sales will be uniformly distributed between 400 and 700 units. The company's operations team must decide how many units to stock per week to meet demand without overstocking. If they stock 550 units, find the probability of overstocking. If they want to have less than a 20% chance of overstocking, should they stock more or less than 550 units?

$0.25$ ; stock more

$0.5$ ; stock more

$0.25$ ; stock less

$0.5$ ; stock less.

Verified step by step guidance

Step 1: Understand the problem. Weekly sales are uniformly distributed between 400 and 700 units. This means the probability density function (PDF) is constant within this range. The operations team stocks 550 units, and we need to calculate the probability of overstocking (sales exceeding 550 units). Additionally, we need to determine whether stocking more or less than 550 units would result in less than a 20% chance of overstocking.

Step 2: Define the uniform distribution. For a uniform distribution, the probability density function is given by $ f(x) = \frac{1}{b-a} $, where $ a $ is the minimum value (400 units) and $ b $ is the maximum value (700 units). The total area under the curve is 1, representing the total probability.

Step 3: Calculate the probability of overstocking. To find the probability of sales exceeding 550 units, calculate the area under the PDF from 550 to 700. This is done using the formula for the uniform distribution: $ P(X > 550) = \int_{550}^{700} f(x) dx $. Substitute $ f(x) = \frac{1}{700-400} $ and solve the integral.

Step 4: Determine the stocking level for less than a 20% chance of overstocking. To achieve this, find the value $ x $ such that $ P(X > x) = 0.2 $. Use the formula $ P(X > x) = \frac{700 - x}{700 - 400} $ and solve for $ x $. This will indicate whether the team should stock more or less than 550 units.

Step 5: Interpret the results. Compare the calculated probability of overstocking at 550 units with the threshold of 20%. Based on the calculations, determine whether stocking more or less than 550 units meets the requirement of less than a 20% chance of overstocking.

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