1. + Chapter 6: Random Variables Section 6.1 Discrete and Continuous Random Variables

2. + Activity: Bottled Water vs. Tap Water This activity will give you a chance to discover whether or not you can taste the difference between bottled water and tap water. 1) Before class begins, numbered stations have been prepared with cups of water. You will be given an index card with a station number on it. 2) Go to the corresponding station. Pick up three cups (labeled A, B, and C) and take them back to your seat. 3) Your task is to determine which one of the three cups contains the bottled water. Drink all the water in Cup A first, then the water in Cup B, and finally the water in Cup C. Write down the letter of the cup that you think held the bottled water. Do not discuss your results with any of your classmates yet? 4) While you taste, your teacher will make a chart on the board

3. + Activity: Bottled Water vs. Tap Water 5) When you are told to do so, record the letter of the cup you identified as containing bottled water on the back of your index card and submit it to your teacher. 6) Your teacher will now reveal the truth about the cups of drinking water. How many students in the class identified the bottled water correctly? What of the class is this? 7) Let’s assume that no one in your class can distinguish tap water from bottled water. In that case, students would just be guessing which cup of water tastes different. If so, what’s the probability that an individual student would guess correctly? 8) How many correct identifications would you need to see to be convinced that the students in your class aren’t just guessing? With your classmates, design and carry out a simulation to answer this question. What do you conclude about your class’s ability to distinguish tap water from bottled water?

4. + Discrete and Continuous Random Variables Random Variable and Probability Distribution A probability model describes the possible outcomes of a chance process and the likelihoodthat those outcomes will occur. A numerical variable that describes the outcomes of a chance process is called a random variable. The probability model for a random variable is itsprobability distribution Definition: 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.

5. Example: 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 Value0123 Probability1/83/8 1/8 Interpret the following: P (X > 1) = 7/8 This means that if we were to repeatedly flip 3 coins and record the number of heads, then in the long-run about 87.5% of time we will get 1 or more heads.

6. + 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 arandom variable and assign probabilities toeach one, we have a discrete random variable. Discrete and Continuous Random Variables

7. 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 x i and their probabilities p i : Value: x 1 x 2 x 3 … Probability: p 1 p 2 p 3 … The probabilities p i must satisfy two requirements: 1.Every probability p i 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 p i of the particular values x i that make up the event. Discrete Random Variables and Their Probability Distributions

8. + Discrete Random Variables Even though a discrete random variable takes a fixed set of possible values, the set of possible values can be infinite. For example, if X = the number of rolls of a fair die needed to get a 6, there is no upper limit for the values of X. Meaning theoretically, it could take an infinite amount of rolls to get a 6.

9. + Example: Babies’ Health at Birth In 1952, Dr. Virginia Apgar suggested five criteria for measuring a baby’s health at birth: skin color, heart rate, muscle tone, breathing, andresponse when stimulated. She developed a 0-1-2 scale to rate anewborn on each of the five criteria. A baby’s Apgar score is the sumof the ratings on each of the five scales, which gives a whole-numbervalue from 0 to 10. Apgar scores are still used today to evaluate thehealth of newborns. What Apgar scores are typical? To find out, researchers recorded the Apgar scores of over 2 million newborn babies in a single year?Imagine selecting one of these newborns at random. (That’s ourchance process.) Define the random variable X = Apgar score of arandomly selected baby one minute after birth. The table below givesthe probability distribution for X Value:012345678910 Probability:0.0010.0060.0070.0080.0120.0200.0380.0990.3190.4370.053

10. + Example: Babies’ Health at Birth (a) Show that the probability distribution for X is legitimate. (b) Make a histogram of the probability distribution. Describe what you see. (c) Apgar scores of 7 or higher indicate a healthy baby. What is P ( X ≥ 7)? (a) All probabilities are between 0 and 1 and they add up to 1. This is a legitimate probability distribution. (b) The left-skewed shape of the distribution suggests a randomly selected newborn will have an Apgar score at the high end of the scale. There is a small chance of getting a baby with a score of 5 or lower. (c) P(X ≥ 7) =.908 We’d have a 91 % chance of randomly choosing a healthy baby. Value:012345678910 Probability:0.0010.0060.0070.0080.0120.0200.0380.0990.3190.4370.053

11. + Checkpoint North Carolina State University posts the grade distributions for its courses online. Students in Statistics 101 in a recent semester received 26% As, 42% Bs, 20% Cs, 10% Ds, and 2% Fs. Choose a Statistics 101 student at random. The student’s grade on a four-point scale (with A = 4) is a discrete random variable X with this probability distribution: 1. Say in words what the meaning of P(X > 3) is. What is this probability? Value of X01234 Probability0.020.100.200.420.26 The probability that the student gets either an A or a B.

12. + Checkpoint 2. Write the event “the student got a grade worse than C” in terms of values of the random variable X. What is the probability of this event? 3. Sketch a graph of the probability distribution. Describe what you see (SOCS). P(X < 2) = 0.12

13. + Mean of a Discrete Random Variable When analyzing discrete random variables, we’ll follow the same strategy we used withquantitative 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, witheach outcome weighted by its probability. Discrete and Continuous Random Variables

14. Definition: 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:

15. + Example: Apgar Scores – What’s Typical? Consider the random variable X = Apgar Score Compute the mean of the random variable X and interpret it in context. Value:012345678910 Probability:0.0010.0060.0070.0080.0120.0200.0380.0990.3190.4370.053 The mean Apgar score of a randomly selected newborn is 8.128. This is the long- term average Apgar score of many, many randomly chosen babies. Note: The expected value does not need to be a possible value of X or an integer! It is a long-term average over many repetitions.

16. + Standard Deviation of a Discrete Random Variable Since we use the mean as the measure of center for a discrete random variable, we’ll use thestandard deviation as our measure of spread.The definition of the variance of a random variable is similar to the definition of the variance for a set of quantitative data. Discrete and Continuous Random Variables

17. Definition: 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 … and that µ X is the mean of X. The variance of X is To get the standard deviation of a random variable, take the square root of the variance.

18. + Example: Apgar Scores – How Variable Are They? Consider the random variable X = Apgar Score Compute the standard deviation of the random variable X and interpret it in context. Value:012345678910 Probability:0.0010.0060.0070.0080.0120.0200.0380.0990.3190.4370.053 The standard deviation of X is 1.437. On average, a randomly selected baby’s Apgar score will differ from the mean 8.128 by about 1.4 units. Variance

19. + AP Exam Tip When asked to calculate the mean (expected value) or standard deviation of a random variable, students lose credit for not showing adequate work or for showing no work at all. To get full credit, students need to go beyond just reporting the commands used on a graphing calculator. In this case, showing the first couple of terms with an ellipsis(…) for the calculation of the mean or standard deviation is sufficient.

20. + Checkpoint A large auto dealership keeps track of sales made during each hour of the day. Let X = the number of cars sold during the first hour of business on a randomly selected Friday. Based on previous records, the probability distribution of X is as follows: 1. Compute and interpret the mean of X. Cars sold:0123 Probability0.30.40.20.1 µ x = 1.1. The long-run average, over many Friday mornings, will be about 1.1 cars sold.

21. + Checkpoint A large auto dealership keeps track of sales made during each hour of the day. Let X = the number of cars sold during the first hour of business on a randomly selected Friday. Based on previous records, the probability distribution of X is as follows: 2. Compute and interpret the standard deviation of X. Cars sold:0123 Probability0.30.40.20.1 σ x = 0.943. On average, the number of cars sold on a randomly selected Friday will differ from the mean (1.1) by 0.943 cars sold.

22. + Continuous Random Variables Discrete random variables commonly arise from situations that involve counting something. Situationsthat involve measuring something often result in a continuous random variable. Discrete and Continuous Random Variables Definition: A continuous random variable X takes on all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event.

23. The probability model of a discrete random variable X assigns a probability between 0 and 1 to each possible value of X. A continuous random variable Y has infinitely many possible values. All continuous probability models assign probability 0 to every individual outcome. Only intervals of values have positive probability.

24. + Some examples will clarify the difference between discrete and continuous variables. Suppose the fire department mandates that all fire fighters must weigh between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire fighter's weight could take on any value between 150 and 250 pounds. Suppose we flip a coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. However, it could not be any number between 0 and plus infinity. We could not, for example, get 2.5 heads. Therefore, the number of heads must be a discrete variable.

25. + Example: Young Women’s Heights The heights of young women closely follow the Normal distribution with mean µ = 64 inches and standarddeviation σ = 2.7 inches. This is a distribution for a large set of data. Now choose one young woman at random.Call her height Y. If we repeat the random choice verymany times, the distribution of values of Y is the sameNormal distribution that describes the heights of all youngwomen. Find the probability that the chosen woman isbetween 68 and 70 inches tall.

26. + Example: Young Women’s Heights Read the example on page 351. Define Y as the height of a randomly chosen young woman. Y is a continuous random variable whose probability distribution is N (64, 2.7). What is the probability that a randomly chosen young woman has height between 68 and 70 inches? P(68 ≤ Y ≤ 70) = ??? P(1.48 ≤ Z ≤ 2.22) = P(Z ≤ 2.22) – P(Z ≤ 1.48) = 0.9868 – 0.9306 = 0.0562 There is about a 5.6% chance that a randomly chosen young woman has a height between 68 and 70 inches.

27. + AP Exam Tip When solving problems involving random variables, start by defining the random variable of interest. For example, let X = the Apgar score of a randomly selected baby or let Y = the height of a randomly selected young woman. Then state the probability you’re trying to find in terms of the random variable: P(68 7).

28. + Section 6.1 Discrete and Continuous Random Variables In this section, we learned that… A random variable is a variable taking numerical values determined by the outcome of a chance process. The probability distribution of a random variable X tells us what the possible values of X are and how probabilities are assigned to those values. A discrete random variable has a fixed set of possible values with gaps between them. The probability distribution assigns each of these values a probability between 0 and 1 such that the sum of all the probabilities is exactly 1. A continuous random variable takes all values in some interval of numbers. A density curve describes the probability distribution of a continuous random variable. Summary

29. + Section 6.1 Discrete and Continuous Random Variables In this section, we learned that… The mean of a random variable is the long-run average value of the variable after many repetitions of the chance process. It is also known as the expected value of the random variable. The expected value of a discrete random variable X is The variance of a random variable is the average squared deviation of the values of the variable from their mean. The standard deviation is the square root of the variance. For a discrete random variable X, Summary