Name: _____________________________________ Section: _______ STA 301 Online Module 6 Activity: Named Discrete and Continuous Distributions Worksheet
1. Recall the pdf for a uniform random variable: 𝑓(𝑥) = Prove that 𝜇 = 𝐸[𝑋] = 1 𝑏−𝑎 for 𝑎 ≤ 𝑥 ≤ 𝑏 𝑎+𝑏 2 2. Of all the new vehicles of a certain model that are sold, 20% require repairs to be done under warranty during the first year of service. A particular dealership sells 12 such vehicles. Let X = the number of vehicles (out of 12) that require warranty repairs Source: Navidi, M. (2008). Statistics for Engineers and Scientists. New York, NY: McGraw-Hill. Explain how you know that X has a binomial distribution by stating and checking all necessary conditions. Provide the values of all parameters of the probability distribution for X. Find 𝐸[𝑋] and 𝑉𝑎𝑟[𝑋]. Find the following binomial probabilities. Be sure to report all relevant R Code as well as the final answer. What is the probability that exactly five of them require warranty repairs? What is the probability that fewer than four of them require warranty repairs? What is the probability that two or more of the them require warranty repairs? Module 6: Named Discrete and Continuous Distributions Page 1
3. The daily high temperature in a computer server room at the university can be modeled by a normal distribution with mean 68.7oF and standard deviation 1.2oF. Source: Johnson, R.A. (2017). Miller & Freund’s Probability and Statistics for Engineers. Boston, MA: Pearson. Directions: For each of the following questions, be sure to sketch and label a distribution and include all relevant R Code. What is the probability that the high temperature on a given day will be between 68.3oF and 70.3oF? Find a symmetric interval about the mean that captures the middle 95% of high temperatures on a given day. Find the quartiles for the high temperature on a given day. Find the 55th percentile for the high temperature on a given day.
4. A bus arrives every 15 minutes at a bus stop. It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution. What is the probability that an individual waits between 3 and 8 minutes? Source: Walpole, R.E., Myers, R.H., Myers, S.L., & Ye, K. (2012). Probability & Statistics for Engineers & Scientists. Boston, MA: Pearson. Module 6: Named Discrete and Continuous Distributions Page 2 • Module 6 Video #1: Distribution Shapes, Bell Curves, and the Empirical Rule (09:47) • Module 6 Video #2: Bernoulli Trials and Introduction to the Binomial Distribution (09:02) • Module 6 Video #3: The Binomial Distribution (15:40) • Module 6 Video #4: The Uniform Distribution (05:49) • Module 6 Video #5: The Normal Distribution Part 1 (09:40) Module 6 Video #6: The Normal Distribution Part 2 (06:57)
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