Chapter 6: Random Variables Section 6.1 Discrete and Continuous Random Variables The Practice of Statistics, 4th edition – For AP* STARNES, YATES, MOORE

Before Test Put phones in holder & orange books on the shelf Late flash cards & reading guides go on the cart in the “late tray”. Remember, any DIGITAL late work requires a full size note in the tray to let me know. Put Ch 5 Flash Cards in the container at the front of the room. Pass Ch 5 Reading Guides to the front. After Test Read pg 339-346 Reading Guide #1-11 Video 6.1 (5 min) & 6.1a (9 min) Estimated test date is Friday March 4

Go ahead & start your reading & vocabulary as soon as you finish your test. Read pg 339-346 Reading Guide #1-11 Video 6.1 (5 min) & 6.1a (9 min) Estimated test date is Friday March 4

Chapter 6 Random Variables Warm-up Open your Socrative app for today’s warm up. Read the directions CAREFULLY Late flash cards & reading guides go on the cart in the “late tray”. Remember, any DIGITAL late work requires a full size note in the tray to let me know.

Chapter 6 Random Variables Warm-up Page 353 #2 a) List the Sample Space and find the probability of each outcome.

Page 353 #2 Value: 2 3 4 5 6 7 8 9 10 11 12 Probability:

Chapter 6 Random Variables 6.1 Discrete and Continuous Random Variables 6.2 Transforming and Combining Random Variables 6.3 Binomial and Geometric Random Variables

Open the “NearPod” app on your tablet. Enter code NCIAH

Discrete and Continuous Random Variables Random Variable and Probability Distribution A probability model describes the possible outcomes of a chance process and the likelihood that 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 its probability distribution Discrete and Continuous Random Variables 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. 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 Value 1 2 3 Probability 1/8 3/8

Discrete and Continuous Random Variables 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 and Continuous Random Variables 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: x1 x2 x3 … Probability: p1 p2 p3 … The probabilities pi must satisfy two requirements: Every probability pi is a number between 0 and 1. 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.

Example: Babies’ Health at Birth page 343 Read the example on page 343. Show that the probability distribution for X is legitimate. Make a histogram of the probability distribution. Describe what you see. Apgar scores of 7 or higher indicate a healthy baby. What is P(X ≥ 7)? Value: 1 2 3 4 5 6 7 8 9 10 Probability: 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053 (a) All probabilities are between 0 and 1 and they add up to 1. This is a legitimate probability distribution. (c) P(X ≥ 7) = .908 We’d have a 91 % chance of randomly choosing a healthy baby. (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.

Page 353 #2 Value: 2 3 4 5 6 7 8 9 10 11 12 Probability: b) The histogram shows that the distribution is symmetric about a center of 7. The values range from 2 to 12. About five-sixths of the time, when you roll a pair of 6-sided dice, you will have a sum of 5 or more. c) P(T>5) = 5/6.

a) The event {X<2} or {X<3} P(X<3) = P(X<2) = 0.49 Page 353 #4 b) The child plays with more than three toys. P(X>3) = 0.28 P(X>3) = 0.51 Number of toys, xi: 1 2 3 4 5 Probability, pi: 0.03 0.16 0.30 0.23 0.17 0.11

a) All the probabilities are between 0 & 1 and they sum to 1. Page 354 #6 a) All the probabilities are between 0 & 1 and they sum to 1. out (had an X-value greater than 0) is mound-shaped and skewed to the right and has a peak at 3 days. b) Most people did not work out in the last 7 days. However, the distribution of those who did work Value: 1 2 3 4 5 6 7 Probability: 0.68 0.05 0.07 0.08 0.04 0.01 0.02 c) The person did not work out all 7 days. P(Y<7) = .98 d) {Y>1}; P(Y>1) = .32

a) A = {1, 2, 3, 4, 5, 6, 7} P(A) = .32 b) B = {0, 1, 2, 3, 4} P(B) = .93 Page 354 #8 c) (AB) = {1, 2, 3, 4} P(AB) = .25; but P(A)*P(B) = .2976 Value: 1 2 3 4 5 6 7 Probability: 0.68 0.05 0.07 0.08 0.04 0.01 0.02 These 2 probabilities are not equal because A and B are not independent.

Discrete and Continuous Random Variables Mean of a Discrete Random Variable When analyzing discrete random variables, we’ll 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. Discrete and Continuous Random Variables Definition: Suppose that X is a discrete random variable whose probability distribution is Value: x1 x2 x3 … Probability: p1 p2 p3 … To find the mean (expected value) of X, multiply each possible value by its probability, then add all the products:

Example: Apgar Scores – What’s Typical? Page 346 Consider the random variable X = Apgar Score Compute the mean of the random variable X and interpret it in context. Value: 1 2 3 4 5 6 7 8 9 10 Probability: 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053 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.

b) E(Y) = µy = ($300)(.9998) + (-$199700)(.0002) = $260 On average, the company gains $260 per policy in the long run. Page 354 #10 Value, yi: $300 -$199,700 Probability, pi: 0.9998 0.0002 E(X) = µX = 2.68 On average, when children are given 5 toys to play with, they play with 2.68 toys. #12 Number of toys, xi: 1 2 3 4 5 Probability, pi: 0.03 0.16 0.30 0.23 0.17 0.11

The company makes an average of $303.35 per life insurance policy. 21 c) E(Y) = µy = Σyipi = $303.35 Page 354 #14 The company makes an average of $303.35 per life insurance policy. Death Age: 21 22 23 24 25 >26 Profit: -$99750 -$99500 Probability: 0.00183 0.00186 0.00189 0.00191 0.00193 Death Age: 21 22 23 24 25 >26 Profit: -$99750 -$99500 -$99250 -99000 -98750 1250 Probability: 0.00183 0.00186 0.00189 0.00191 0.00193 .99058

Discrete and Continuous Random Variables 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 the standard 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 Definition: Suppose that X is a discrete random variable whose probability distribution is Value: x1 x2 x3 … Probability: p1 p2 p3 … and that µX 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.

Example: Apgar Scores – How Variable Are They? page 347 Consider the random variable X = Apgar Score Compute the standard deviation of the random variable X and interpret it in context. Value: 1 2 3 4 5 6 7 8 9 10 Probability: 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053 Variance 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.

X = 1.3106 This means that, on average, the number of toys a randomly selected child will play with will differ from the mean (2.68) by 1.31 toys. Page 355 #16 Number of toys, xi: 1 2 3 4 5 Probability, pi: 0.03 0.16 0.30 0.23 0.17 0.11

Page 355 #18 a) Even though the company will lose a large amount of money on a small number of policyholders who die, it will gain a small amount from many thousands of 21-year-old men. In the long run, the insurance company can expect to make $303.35 per insurance policy, on average. Death Age: 21 22 23 24 25 >26 Profit: -$99750 -$99500 -$99250 -99000 -98750 1250 Probability: 0.00183 0.00186 0.00189 0.00191 0.00193 .99058 b) Y = $9707.57

Read pg 346-353 Do pg 353 #1-13 odd Reading Guide #12-20 Chapter 6 Vocabulary Flash Cards Estimated Chapter 6 Test Date is Friday March 4.