Random Variables Lesson 5 1 Two Types of

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

Random Variables • A random variable assumes a value based on the outcome of a random event. – We use a capital letter, like X, to denote a random variable. – A particular value of a random variable will be denoted with the corresponding lower case letter, in this case x. Slide 16 - 3

Random Variables • There are two types of random variables: – Discrete random variables can take one of a countable number of distinct outcomes. • Example: Number of M&M’s in a bag – Continuous random variables can take any numeric value within a range of values. • Example: Weights of Jets Defensive Team Slide 16 - 4

TWO TYPES OF RANDOM VARIABLES RANDOM VARIABLE ARE LIKE THE WATER COMING OUT OF A FAUCET DISCRETE RANDOM VARIABLES ARE LIKE THE DRIPS OF WATER, INDIVIDUAL DRIPS WITH GAPS IN BETWEEN CONTINUOUS RANDOM VARIABLES ARE LIKE THE FLOWING WATER Statistics and Probability with Applications, 3 rd Edition 5

Random Variables • A probability model for a random variable consists of: – The collection of all possible values of a random variable, and – the probabilities that the values occur. Slide 16 - 6

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: Value 0 1 2 3 Probability 1/8 3/8 1/8 Statistics and Probability with Applications, 3 rd Edition 7

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 8

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: Value x 1 x 2 x 3 … Probability p 1 p 2 p 3 … 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 9

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: Value 0 1 2 3 Probability 1/8 3/8 1/8 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 10

Finding probabilities with discrete random variables A valid probability distribution for the 2015 AP Statistics exam is shown below, where X = the exam score of a randomly selected student. A score of 3 or more is considered a passing score. What is the probability that a randomly selected student has a passing score? Score Probability 1 2 3 4 5 0. 236 0. 186 0. 252 0. 191 0. 135 Statistics and Probability with Applications, 3 rd Edition 11

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 12

Discrete or continuous • PROBLEM: Suppose a professional tennis player hires a statistician to record data on all of her serves in matches. By the end of her career, she has data for thousands of serves. Suppose the tennis player randomly selects one of those serves. Classify each of the following random variables as discrete or continuous. • (a) S = the speed (in mph) of the serve. • (b) B = 0 if the serve landed out of bounds and 1 if the serve landed in bounds. • (c) R = the length of the rally (the number of shots until the point is over) resulting from the serve. Statistics and Probability with Applications, 3 rd Edition 13

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. Value 0 1 2 3 4 Probability 0. 011 0. 032 0. 138 0. 362 0. 457 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 14

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 15