Chapter 7: Random Variables

1. Chapter 7: Random Variables

2. Jakob Bernoulli • The Law of Large Numbers – Read the preface to Chapter 7 on page 388 and be prepared to summarize the Law of Large Numbers.

3. Activity 7 • Break into pairs and go through Activity 7 on page 390. Use the Random Number Generator on your calculator to generate rolls of two dice for this activity. • RanInt(2,12) • Be prepared to share data with the class so we can put all the data together.

4. Activity 7 • 1. How many times out of 20 games did your pair win at craps? Relative frequency wins divided by 20. • 2. Combined data. Relative frequency of wins for the class. • 3. Repeat the process in number 1 but play 25 games. Relative win frequency for the class when 25 games are played. Compared to answer in number 2.

5. Activity 7 • 4. Relative frequency on all 45 rolls of starting a game and winning with a 7 or 11 for each group. Combined relative frequency for the class. • 5. Relative frequency of crapping out on the first roll for all 45 attempts for each group. Relative frequency for the class of crapping out on the first roll.

6. Activity 7 • 6. Ok simulate 36 rolls of the dice and record the relative frequency for each number. • Number of 2’s divided by 36, number of 3’s divided by 36, … , number of 12’s divided by 36 for each group. Combined class results. • 7. Create a histogram from 6. Shape of the distribution, center of the distribution, least likely to occur, …

7. Activity 7 • Using your histogram from number 6, what is the relative frequency of winning on the first roll and the relative frequency of losing on the first roll? How do these compare with the relative frequencies from our simulations?

8. Random Variable • A random variable is a variable whose value is a numerical outcome of a random phenomenon. • How could we express tossing a coin four times as a random variable?

9. 7.1 Discrete and Continuous 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. • See page 392. • The sum of these probabilities sh0uld be 1. • The probability of each event of X must have a probability between 0 and 1.

10. Example 7.1 • Read and discuss with a neighbor. Be able to discuss with the class.

11. Probability histogram- a histogram that displays the probabilities of the outcomes of a phenomenon. • See page 393.

12. Example 7.2 • Read and discuss with a neighbor. Be prepared to discuss with the class. • Assignment 7.1 through 7.5 problems starting on page 395.

13. Continuous Random Variables Let S = { all numbers such that 0 < x < 1} If the spinner in figure 7.4 is spun what is the probability of spinning a given number? What does the distribution look like?

15. Example 7.3 • Read and Discuss with a neighbor.

17. Continuous Random Variable • A continuous random variable X takes all values in an interval of numbers. • The probability distribution of X is described as a density curve.

19. All continuous probability distributions assign probability 0 to every individual event.

20. Normal Distributions as Probability Distributions • Z scores? What is a z-score? • What is the formula to convert an x-score to a z-score?

21. Example 7.4 • Read and be ready to discuss with a neighbor.

22. 7.2 Means and Variances of Random Variables • The mean of a random variables X is also an average of the possible values of x, but with an essential change to take into account the fact that not all outcomes need be equally likely. • The expected value is the mean of a random variable. The expected value is a weighted average.

23. Mean of a discrete variable

24. Example 7.5 • Read and Discuss with a neighbor, be prepared to discuss with the class.

25. Example 7.6 • Read and Discuss with a neighbor, be prepared to discuss with the class.

26. Variance of a Random Variable • The variance of a variable is the squared value of the standard deviation.

28. Example 7.7 • Read and Discuss with a neighbor, be prepared to discuss with the class.

29. Law of Large Numbers • Pg.413 • We can get a statistic really close to a parameter by doing larger and larger samples. The larger the sample the closer we will get to the paremeter.

30. Example 7.8 • Read and Discuss with a neighbor, be prepared to discuss with the class.

31. Law of Small Numbers • People believe that a small run of events should behave like a large run of the events. This is NOT true. Even though Shaquille O’Neal’s lifetime free throw percentage was dismal. He did have runs of getting 5 and 6 free throws in at a time successfully.

32. Example 7.9 • Read and Discuss with a neighbor, be prepared to discuss with the class.

33. How large of a number is required? • The larger the variability of an event the more trials required to ensure the statistic is close to the parameter.

34. Rules for Means • Rule 1: If X is a random variable and a and b are fixed numbers, then

35. Rules for Means • Rule 2: If X and Y are random variables, then

36. Example 7.10 • Read and Discuss with a neighbor, be prepared to discuss with the class.

37. Rules for Variances • Rule 1. If X is a random variable and a and b are fixed numbers, then

38. Rules for Variances • Rule 2. If X and Y are independent random variables, then • This is the addition rule for variances of independent random variables.

39. Rules for Variances • Rule 3. If X and Y have correlation ρ, then • This is the general rule for variances of random variables. The correlation between two independent events is zero.

40. Example 7.11 • Read and Discuss with a neighbor, be prepared to discuss with the class.

41. Example 7.12 • Read and Discuss with a neighbor, be prepared to discuss with the class.

42. Example 7.13 • Read and Discuss with a neighbor, be prepared to discuss with the class.

43. Example 7.14 • Read and Discuss with a neighbor, be prepared to discuss with the class. • Zny linear combination of independent random variables is also normally distributed.