5 Random Variables Lesson 5 1 Two Types

5 Random Variables Lesson 5. 1 Two Types of Random Variables Statistics and Probability with Applications, 3 rd Edition Starnes & Tabor Bedford Freeman Worth Publishers

Two Types of Random Variables Learning Targets After this lesson, you should be able to: ü Verify that the probability distribution of a discrete random variable is valid. ü Calculate probabilities involving a discrete random variable. ü Classify a random variable as discrete or continuous. Statistics and Probability with Applications, 3 rd Edition 2

Two Types of Random Variables Suppose you toss a fair coin 3 times. The sample space for this chance process is HHH HHT HTH THH HTT THT TTH TTT Because there are 8 equally likely outcomes, the probability is 1/8 for each possible outcome. Define the random variable X = the number of heads obtained in 3 tosses. The value of X will vary from one set of tosses to another, but it will always be one of the numbers 0, 1, 2, or 3. How likely is X to take each of those values? X=0: TTT X=1: HTT THT TTH X=2: HHT HTH THH X=3: HHH We can summarize the probability distribution of X in a table: Statistics and Probability with Applications, 3 rd Edition 3

Two Types of Random Variables Random Variable, Probability Distribution A random variable takes numerical values that describe the outcomes of a chance process. The probability distribution of a random variable gives its possible values and their probabilities. There are two main types of probability distributions, corresponding to two types of random variables: discrete and continuous. Statistics and Probability with Applications, 3 rd Edition 4

Two Types of Random Variables Discrete Random Variable A discrete random variable X takes a fixed set of possible values with gaps between. The probability distribution of a discrete random variable X lists the values xi and their probabilities pi: In a valid probability distribution, the probabilities pi must satisfy two requirements: 1. Every probability pi is a number between 0 and 1, inclusive. 2. The sum of the probabilities is 1: p 1 + p 2 + +. . . = 1. Statistics and Probability with Applications, 3 rd Edition 5

Two Types of Random Variables We can use the probability distribution of a discrete random variable X to find the probability of an event. The probability distribution for X = the number of heads obtained when tossing a fair coin 3 times is: What’s the probability that we get at least one head in 3 tosses of the coin? P(X ≥ 1) = P(X=1 or X=2 or X=3) Because the events X = 1, X = 2, and X = 3 are mutually exclusive, we can add their probabilities to get the answer: P(X ≥ 1) = P(X=1) + P(X=2) + P(X=3) = 1/8 + 3/8 = 7/8 Or we could use the complement rule from Lesson 4. 2: P(X ≥ 1) =1 − P(X < 1) = 1 − P(X=0) = 1 − 1/8 = 7/8 Statistics and Probability with Applications, 3 rd Edition 6

Two Types of Random Variables Suppose we want to randomly select a number between 0 and 1, allowing any number between 0 and 1 as the outcome (like 0. 84522 or 0. 1111119). The sample space of this chance process is the entire interval of values between 0 and 1 on a number line. If we define Y = the outcome of the random number generator, then Y is a continuous random variable. Continuous Random Variable A continuous random variable X can take any value in an interval on the number line. • Most discrete random variables result from counting something. • Continuous random variables typically result from measuring something. Statistics and Probability with Applications, 3 rd Edition 7

LESSON APP 5. 1 Making the grade? Indiana University Bloomington posts the grade distributions for its courses online. In a recent semester of a Business Statistics course, 45. 7% of students received A’s, 36. 2% B’s, 13. 8% C’s, 3. 2% D’s, and 1. 1% F’s. Choose a Business Statistics student at random. The student’s grade on a 4 -point scale (with A = 4) is a random variable X with this probability distribution: 1. 2. 3. 4. Is X a discrete or a continuous random variable? Explain. Show that the probability distribution of X is valid. Explain in words what P(X ≥ 3) means. What is this probability? Write the event “the student got a grade worse than C” using probability notation. What is the probability of this event? Statistics and Probability with Applications, 3 rd Edition 8

Two Types of Random Variables Learning Targets After this lesson, you should be able to: ü Verify that the probability distribution of a discrete random variable is valid. ü Calculate probabilities involving a discrete random variable. ü Classify a random variable as discrete or continuous. Statistics and Probability with Applications, 3 rd Edition 9