Activity 11

MATH 216: Statistical Thinking

Activity 1: Think-Pair-Share

  1. Think: Reflect on the following question:

    • Why is the probability of a continuous random variable taking on a specific value always zero?
    • How does the area under the PDF curve relate to probability?

  2. Pair: Discuss your thoughts with a partner and come up with a joint explanation.

  3. Share: Share your insights with the class, focusing on how the properties of PDFs differ from those of discrete probability mass functions.

Activity 2: Quiz

  • Question: Suppose \(X\) is uniformly distributed between \(a = 2\) and \(b = 8\).

    • What is the probability that \(X\) falls between 3 and 5?
    • Calculate the mean and variance of \(X\).

Instructions: Use the formula for the uniform distribution:

  • \(P(c \leq X \leq d) = \frac{d-c}{b-a}\)
  • \(\mu = \frac{a+b}{2}\)
  • \(\sigma^2 = \frac{(b-a)^2}{12}\)

Activity 3: Data Exploration

  • Modify the values of \(a\) and \(b\) to see how the distribution changes.
  • Calculate the mean and variance of the generated data and compare them to the theoretical values.

Activity 4: Probability Calculations

Prompt: An unprincipled used-car dealer sells a car to an unsuspecting buyer, even though the dealer knows that the car will have a major breakdown within the next 6 months. The dealer provides a warranty of 45 days on all cars sold. Let \(x\) represent the length of time until the breakdown occurs. Assume that x is a uniform random variable with values between 0 and 6 months.

  1. Calculate and interpret the mean and standard deviation of \(X\).
  1. Graph the probability distribution of \(X\) , and show the mean on the horizontal axis. Also show one and two-standard-deviation intervals around the mean.
  1. Calculate the probability that the breakdown occurs while the car is still under warranty.