Activity 11

MATH 216: Statistical Thinking

Continuous Random Variables: Uniform Distribution

Time Allocation: 15 minutes total (5 min reading, 10 min individual work)

Part 1: Conceptual Understanding (3 minutes)

Instructions: Answer the following questions about continuous random variables:

  1. Why is the probability of a continuous random variable taking on a specific value always zero?
  1. How does the area under a probability density function (PDF) curve relate to probability?
  1. What is the key difference between probability mass functions (for discrete variables) and probability density functions (for continuous variables)?

Part 2: Uniform Distribution Calculations (4 minutes)

Scenario: Suppose X is uniformly distributed between a = 2 and b = 8.

Instructions: Use the uniform distribution formulas to calculate:

  • \(P(3 \leq X \leq 5)\) = \((d-c)/(b-a)\) =

  • Mean (\(\mu\)) = \((a+b)/2\) =

  • Variance (\(\sigma^2\)) = \((b-a)^2/12\) =

  • Standard Deviation (\(\sigma\)) = \(\sqrt{\sigma^2}\) =

Show your work for at least two calculations:

Part 3: Real-World Application (3 minutes)

Scenario: An unprincipled used-car dealer sells a car with a known major breakdown risk within the next 6 months. The dealer provides a 45-day warranty. Let \(X\) represent the time until breakdown (in months), uniformly distributed between 0 and 6 months.

Calculate:

  • Mean time until breakdown: months
  • Standard deviation: months
  • Probability breakdown occurs during warranty (45 days = 1.5 months):

Critical Thinking: Why might this uniform distribution assumption be unrealistic for car breakdown times?