Lesson Probability Distributions of Continuous Random Variables

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1 STATWAY STUDENT HANDOUT STUDENT NAME DATE INTRODUCTION In previous lessons, you organized outcomes and probabilities in a probability distribution. This works well when the possible outcomes can actually be listed. The outcomes used were all discrete; this means that you can graph all outcomes as individual points on a number line. You could actually list all outcomes. What if that is not possible? Let s say the data are a collection of the heights of freshmen at your college. Depending on the accuracy of the measurement tool, there are potentially an infinite number of possible heights. You could report the results to the nearest inch, half inch, quarter inch, eighth inch, and so on. These are not discrete data. You cannot possibly list all possible heights. These data are continuous. TRY THESE PART 1 1 Classify each of the random variables described as either discrete or continuous. A The time it takes to run a mile. B The number of problems assigned for homework. C The number of shoes in a closet. D The lifetime of an automobile tire. E The circumference of a hailstone. F The length of a giraffe s neck. G The sum of the roll of two dice.

2 STATWAY STUDENT HANDOUT 2 NEXT STEPS Because continuous variables are measurements we need a way to represent all possibilities for a continuous data set. A number line or horizontal axis can be used to represent all the possibilities since the infinite values are all points on a number line. The height of the graph at any point represents the relative frequency of that point. The total area under the curve is equal to 1 since the sum of all probabilities in a probability distribution is 1. This follows from your work with discrete random variables where the sum of all probabilities is equal to 1. Suppose you define a continuous random variable as the time it takes to pack a customer order for shipping at a particular company and the results are uniformly distributed between 5 and 15 minutes, (i.e., the probability for each packing time is the same). The distribution is represented by the graph below. If the area of the rectangle is 1, what would be the height of the rectangle? (label the vertical axis)

3 STATWAY STUDENT HANDOUT 3 TRY THESE PART 2 2 What is the probability that an order takes 15 minutes or less to pack? 3 What is the probability that an order takes 12 minutes or more to pack? 4 What is the probability that an order takes between 9 and 13 minutes to pack? 5 What is the probability that an order takes less than 8 minutes or more than 11 minutes to pack? 6 What is the probability that an order takes more than 25 minutes to pack? 7 If the supervisor wants to study the 20% of orders taking the longest time to pack, he needs to study orders taking at least how long?

4 STATWAY STUDENT HANDOUT 4 TAKE IT HOME 1 A computer program randomly selects real numbers between 0 and 25. If the program is truly random, the results should be uniformly distributed. Sketch the distribution, shade the appropriate area, and find the probability for each of the following questions. A What is the probability of randomly selecting a real number between 5 and 10? B What is the probability of randomly selecting a real number of at least 16? C What is the probability of randomly selecting a real number not larger than 20?? D At what number are 30% of the randomly selected real numbers below? E What is the probability of randomly selecting a real number less than 0?? F What is the probability of randomly selecting a real number greater than 20 or less than 5??

5 STATWAY STUDENT HANDOUT 5 2 Pecan trees drop an organic compound called juglone that inhibits growth of new trees and many other plants. This naturally controls overcrowding. A pecan farmer is interested in finding the optimal distance between any healthy mature pecan tree to the next closest healthy tree. Planting trees too close together compromises pecan production, while planting trees too far apart sacrifices potential income. If he hopes to have 80% of the trees grow to healthy maturity, how far apart should he plant them? Assume the distribution of distances between healthy trees follows an approximately uniform distribution of feet This lesson is part of STATWAY, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, A Pathway Through Statistics, Statway was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel.

6 STATWAY STUDENT HANDOUT STATWAY and the Carnegie Foundation logo are trademarks of the Carnegie Foundation for the Advancement of Teaching. A Pathway Through College Statistics may be used as provided in the CC BY license, but neither the Statway trademark nor the Carnegie Foundation logo may be used without the prior written consent of the Carnegie Foundation.