Sec$on 6.1: Discrete and Con.nuous Random Variables. Tuesday, November 14 th, 2017

Transcription

1 Sec$on 6.1: Discrete and Con.nuous Random Variables Tuesday, November 14 th, 2017

2 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 and standard deviation of a discrete random variable. Thursday: ü COMPUTE probabilities using the probability distribution of certain continuous random variables. The Practice of Statistics, 5 th Edition 2

3 Introduc$on: Suppose we toss a fair coin 3 $mes. 1. What is the sample space for this chance process? How many outcomes are possible? What is the likelihood of each outcome? Sample Space: HHH HHT HTH HTT THH THT TTH TTT There are 8 equally likely outcomes in our sample space. The probability is 1/8 for each outcome.

4 2. Let X = number of heads obtained in a toss. The value of X will vary from,,, or. How likely is X to take each of those values?

5 3. Find and interpret P(X 1). This probability means that if we were to repeatedly flip 3 coins and record the number of heads, we will get 1 or more heads about 87.5% of the $me.

6 What is a random variable? What are the two main types of random variables? A numerical value that describes the outcomes of a chance process. DISCRETE RANDOM VARIABLES CONTINUOUS RANDOM VARIABLES

7 What is a probability distribu.on? A probability distribu$on is a model that gives the possible values and probabili$es of a random variable. Can be given as tables or histograms.

8 What is a discrete random variable? Give some examples. A random variable with gaps between the values Discrete random variables are o=en>mes things you can count. For example, shoe size vs. actual length of foot.. Which one is discrete?

9 EXAMPLE 1: Apgar Scores (Babies Health at Birth)

10 a) Show the probability model is legi.mate. All of the probabili.es for the Apgar scores are between 0 and 1. The probabili.es add up to 1. Therefore, we can conclude that this is a legi.mate probability model.

11 b) Make a histogram of the probability distribu.on. Describe what you see. Shape: the distribu.on appears skewed lex and single-pinked. A randomly selected newborn is most likely to have a higher Apgar score, meaning they are healthy. Center: The median Apgar score appears to be a score of 8. Spread: Apgar scores vary between 0 and 10, although most of the scores are between 6 and 10.

12 c) Doctors decided that Apgar scores of 7 or higher indicate a healthy baby. What s the probability that a randomly selected baby is healthy? We would have about a 91% chance of selec$ng a healthy newborn.

13 READ P

14 Mean (Expected Value) of a Discrete Random Variable 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: µ x = E X ) = x p + x p + x p +... = x ( i p i

15 d) 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 This is the average Apgar score of many, many randomly chosen babies.

16 PG

17 Standard Deviation of a Discrete Random Variable The definition of the variance of a random variable is similar to the definition of the variance for a set of quantitative data. 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 Var(X) = σ X 2 = (x 1 µ X ) 2 p 1 + (x 2 µ X ) 2 p 2 + (x 3 µ X ) 2 p = (x i µ X ) 2 p i To get the standard deviation of a random variable, take the square root of the variance.

18 e) Compute and interpret the standard devia.on of the random variable X. The standard deviation of X is σ X = (2.066) = A randomly selected baby s Apgar score will typically differ from the mean (8.128) by about 1.4 units.

19 ! Now You Try!! 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.!!! * 5 randomly selected students will write and present their answers to the class on the whiteboard *

20 Show that the probability distribu.on for X is legi.mate. All the probabili.es are between 0 and 1 and they add to 1, so this is a legi.mate probability distribu.on.

21 Make a histogram of the probability distribu.on. 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.

22 (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) = = There is a probability of randomly selec$ng a student who speaks three or more languages.

23 (d) Compute the mean of the random variable X and interpret this value in context.

24 (e) Compute and interpret the standard devia.on of the random variable X.

25 Exit Ticket: Checks for Understanding Complete the following Check Your Understanding Problems on binder paper to turn in: Pg. 350 #1-3 Pg. 355 #1-2

26 By next class: READ about con.nuous random variables and finish the notes.

27 Chapter 5 Quiz: Thursday! HW #21 is due Thursday Complete the Chapter 5 Prac.ce Test on your own Come by to tutorial for extra help!