The Practice of Statistics Fifth Edition 6 1 Discrete and Continuous Random Variables Answers
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CHAPTER 6 Random Variables 6. 1 a Discrete and Continuous Random Variables The Practice of Statistics, 5 th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers
Discrete and Continuous Random Variables Learning Objectives After this section, you should be able to: ü COMPUTE probabilities using the probability distribution of a discrete random variable. ü CALCULATE and INTERPRET the mean (expected value) of a discrete random variable. ü CALCULATE and INTERPRET the standard deviation of a discrete random variable. ü COMPUTE probabilities using the probability distribution of certain continuous random variables. The Practice of Statistics, 5 th Edition 2
Random Variables and Probability Distributions A probability model describes the possible outcomes of a chance process and the likelihood that those outcomes will occur. Consider tossing a fair coin 3 times. Define X = the number of heads obtained X = 0: TTT X = 1: HTT THT TTH X = 2: HHT HTH THH X = 3: HHH Value 0 1 2 3 Probability 1/8 3/8 1/8 A random variable takes numerical values that describe the outcomes of some chance process. The probability distribution of a random variable gives its possible values and their probabilities. Interpret P(X > 1) = 7/8 If we repeatedly flip 3 coins and record the number of heads, we will get 1 or more heads 87. 5% of the time. The Practice of Statistics, 5 th Edition Reminder: "distribution" is the same since Ch 1: the possible values a variable can take and how often it takes those values 3
Discrete Random Variables There are two main types of random variables: discrete and continuous. If we can find a way to list all possible outcomes for a random variable and assign probabilities to each one, we have a discrete random variable. Discrete Random Variables And Their Probability Distributions 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 … The probabilities pi must satisfy two requirements: 1. Every probability pi is a number between 0 and 1. 2. The sum of the probabilities is 1. To find the probability of any event, add the probabilities pi of the particular values xi that make up the event. The Practice of Statistics, 5 th Edition 4
Example - How many languages? Imagine selecting a U. S. high school student at random. Define the random variable X = number of languages spoken by the randomly selected student. The table below gives the probability distribution of X, based on a sample of students from the U. S. Census at School database. Languages: Probability: 1 0. 630 2 0. 295 3 0. 065 4 0. 008 5 0. 002 Problem: (a) Show that the probability distribution for X is legitimate. (b) Make a histogram of the probability distribution. Describe what you see. (c) What is the probability that a randomly selected student speaks at least 3 languages? The Practice of Statistics, 5 th Edition 5
Example - How many languages? Languages: 1 2 3 4 Probability: 0. 630 0. 295 0. 065 0. 008 (a) Show that the probability distribution for X is legitimate. All the probabilities are between 0 and 1 and they add to 1, so this is a legitimate probability distribution. 5 0. 002 (b) Make a histogram of the probability distribution. Describe what you see. Shape: The histogram is strongly skewed to the right. Center: The median is 1, but the mean will be slightly higher due to the skewness. Spread: The number of languages varies from 1 to 5, but nearly all of the students speak just one or two languages. (c) What is the probability that a randomly selected student speaks at least 3 languages? P(X > 3) = P(X = 3) + P(X = 4) + P(X = 5) = 0. 065 + 0. 008 + 0. 002 = 0. 075. There is a 0. 075 probability of randomly selecting a student who speaks three or more languages. The Practice of Statistics, 5 th Edition 6
Example - Roulette On an American roulette wheel, there are 38 slots numbered 1 through 36, plus 0 and 00. Half of the slots from 1 to 36 are red; the other half are black. Both the 0 and 00 slots are green. Suppose that a player places a simple $1 bet on red. If the ball lands in a red slot, the player gets the original dollar back, plus an additional dollar for winning the bet. If the ball lands in a different-colored slot, the player loses the dollar bet to the casino. Let's define the random variable X = net gain from a single $1 bet on red. Determine the probability distribution. Value Probability The Practice of Statistics, 5 th Edition 7
Example - Roulette • Value Probability This means the player loses (and the casino gains) an average of five cents per $1 bet in many, many plays of the game. The Practice of Statistics, 5 th Edition 8
Mean (Expected Value) of a Discrete Random Variable When analyzing discrete random variables, we follow the same strategy we used with quantitative data – describe the shape, center, and spread, and identify any outliers. The mean of any discrete random variable is an average of the possible outcomes, with each outcome weighted by its probability. Suppose that X is a discrete random variable whose probability distribution is Value: x 1 x 2 x 3 … Probability: p 1 p 2 p 3 … To find the mean (expected value) of X, multiply each possible value by its probability, then add all the products: The Practice of Statistics, 5 th Edition 9
Mean (Expected Value) of a Discrete Random Variable A baby's Apgar score is the sum of the ratings on each of five scales, which gives a whole-number value from 0 to 10. Let X = Apgar score of a randomly selected newborn Compute the mean of the random variable X. Interpret this value in context. We see that 1 in every 1000 babies would have an Apgar score of 0; 6 in every 1000 babies would have an Apgar score of 1; and so on. So the mean (expected value) of X is: The mean Apgar score of a randomly selected newborn is 8. 128. This is the average Apgar score of many, many randomly chosen babies. The Practice of Statistics, 5 th Edition 10
Example - How many languages? • Languages: Probability: The Practice of Statistics, 5 th Edition 1 0. 630 2 0. 295 3 0. 065 4 0. 008 5 0. 002 11
Discrete and Continuous Random Variables Section Summary In this section, we learned how to… ü COMPUTE probabilities using the probability distribution of a discrete random variable. ü CALCULATE and INTERPRET the mean (expected value) of a discrete random variable. ü CALCULATE and INTERPRET the standard deviation of a discrete random variable. ü COMPUTE probabilities using the probability distribution of certain continuous random variables. ü Read p. 345 -352 ccc 1, 3, 5, 7, 9, 11, 13 The Practice of Statistics, 5 th Edition 12
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