1. AP Statistics, Section 7.11 The Practice of Statistics Third Edition Chapter 7: Random Variables 7.1 Discete and Continuous Random Variables Copyright © 2008 by W. H. Freeman & Company Daniel S. Yates

2. AP Statistics, Section 7.12 Essential Questions What is discrete random variable? What is a probability distribution? How do you construct a probability distribution for a discrete random variable? Given a probability distribution for a random variable, how do you construct a probability histogram? What is a density curve? What is a uniform distribution? What is a continuous random variable and how do you define a probability distribution for a continuous random variable?

3. AP Statistics, Section 7.13 Random Variables A random variable is a variable whose value is a numerical outcome of a random phenomenon. For example: Flip three coins and let X represent the number of heads. X is a random variable. We usually use capital letters to denotes random variables. The sample space S lists the possible values of the random variable X. We can use a table to show the probability distribution of a discrete random variable.

4. AP Statistics, Section 7.14 Discrete Probability Distribution Table Value of X:x1x1 x2x2 x3x3 …xnxn Probability: p1p1 p2p2 p3p3 …pnpn

5. AP Statistics, Section 7.15 Discrete Random Variables A discrete random variable X has a countable number of possible values. The probability distribution of X lists the values and their probabilities. X: x 1 x 2 x 3 … x k P(X): p 1 p 2 p 3 … p k 1. 0 ≤ p i ≤ 1 2. p 1 + p 2 + p 3 +… + p k = 1.

6. AP Statistics, Section 7.16 Probability Distribution Table: Number of Heads Flipping 4 Coins TTTT TTTH TTHT THTT HTTT TTHH THTH HTTH HTHT THHT HHTT THHH HTHH HHTH HHHT HHHH X01234 P(X)1/164/166/164/161/16

7. AP Statistics, Section 7.17 Probabilities: X: 0 1 2 3 4 P(X): 1/16 1/4 3/8 1/4 1/16.0625.25.375.25.0625 Histogram

8. AP Statistics, Section 7.18 Questions. Using the previous probability distribution for the discrete random variable X that counts for the number of heads in four tosses of a coin. What are the probabilities for the following? P(X = 2) P(X ≥ 2) P(X ≥ 1).375.375 +.25 +.0625 =.6875 1-.0625 =.9375

9. AP Statistics, Section 7.19 What is the average number of heads?

10. AP Statistics, Section 7.110 Continuous Random Varibles Suppose we were to randomly generate a decimal number between 0 and 1. There are infinitely many possible outcomes so we clearly do not have a discrete random variable. How could we make a probability distribution? We will use a density curve, and the probability that an event occurs will be in terms of area.

11. AP Statistics, Section 7.111 Distribution of Continuous Random Variable

12. AP Statistics, Section 7.112 Problem Let X be the amount of time (in minutes) that a particular San Francisco commuter must wait for a BART train. Suppose that the density curve is a uniform distribution. Draw the density curve for 0 to 20 minutes. What is the probability that the wait is between 12 and 20 minutes?

13. AP Statistics, Section 7.113 Density Curve.

14. AP Statistics, Section 7.114 Probability shaded. P(12≤ X ≤ 20) = 0.5 · 8 =.40

15. AP Statistics, Section 7.115 Normal Curves We’ve studied a density curve for a continuous random variable before with the normal distribution. Recall: N(μ, σ) is the normal curve with mean μ and standard deviation σ. If X is a random variable with distribution N(μ, σ), then is N(0, 1)

16. AP Statistics, Section 7.116 Example Students are reluctant to report cheating by other students. A sample survey puts this question to an SRS of 400 undergraduates: “You witness two students cheating on a quiz. Do you go to the professor and report the cheating?” Suppose that if we could ask all undergraduates, 12% would answer “Yes.” The proportion p = 0.12 would be a parameter for the population of all undergraduates.

17. AP Statistics, Section 7.117 Example continued Students are reluctant to report cheating by other students. A sample survey puts this question to an SRS of 400 undergraduates: “You witness two students cheating on a quiz. Do you go to the professor and report the cheating?” What is the probability that the survey results differs from the truth about the population by more than 2 percentage points? Because p = 0.12, the survey misses by more than 2 percentage points if

18. AP Statistics, Section 7.118

19. AP Statistics, Section 7.119 Example continued Calculations About 21% of sample results will be off by more than two percentage points.

20. AP Statistics, Section 7.120 Summary A discrete random variable X has a countable number of possible values. The probability distribution of X lists the values and their probabilities. A continuous random variable X takes 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.

21. AP Statistics, Section 7.121 Summary When you work problems, first identify the variable of interest. X = number of _____ for discrete random variables. X = amount of _____ for continuous random variables.