1. Discrete Probability Distributions
Chapter 6
Discrete Probability Distributions

2. Discrete Random Variables
Section
6.1
Discrete Random Variables

3. Objectives
Distinguish between discrete and continuous random variables
Identify discrete probability distributions
Construct probability histograms
Compute and interpret the mean of a discrete random variable
Interpret the mean of a discrete random
Compute the standard deviation of a discrete random variable 3

4. Objective 1
Distinguish between Discrete and Continuous Random Variables 4

5. A random variable is a numerical measure of the outcome from a probability experiment, so its value is determined by chance. Random variables are denoted using letters such as X. 5

6. A discrete random variable has either a finite or countable number of values. The values of a discrete random variable can be plotted on a number line with space between each point. 6

7. A continuous random variable has infinitely many values
A continuous random variable has infinitely many values. The values of a continuous random variable can be plotted on a line in an uninterrupted fashion. 7

8. (b) The number of leaves on a randomly selected oak tree.
EXAMPLE Distinguishing Between Discrete and Continuous Random Variables
Determine whether the following random variables are discrete or continuous. State possible values for the random variable.
The number of light bulbs that burn out in a room of 10 light bulbs in the next year.
(b) The number of leaves on a randomly selected oak tree.
(c) The length of time between calls to 911.
Discrete; x = 0, 1, 2, …, 10
Discrete; x = 0, 1, 2, …
Continuous; t > 0 8

9. Objective 2
Identify Discrete Probability Distributions 9

10. A probability distribution provides the possible values of the random variable X and their corresponding probabilities. A probability distribution can be in the form of a table, graph or mathematical formula. 10

11. EXAMPLE A Discrete Probability Distribution
The table to the right shows the probability distribution for the random variable X, where X represents the number of movies streamed on Netflix each month. x
P(x)
0.06 1
0.58 2
0.22 3
0.10 4
0.03 5
0.01 11

12. Rules for a Discrete Probability Distribution
Let P(x) denote the probability that the random variable X equals x; then
Σ P(x) = 1
0 ≤ P(x) ≤ 1 12

13. Is the following a probability distribution?
EXAMPLE Identifying Probability Distributions
Is the following a probability distribution? x
P(x)
0.16 1
0.18 2
0.22 3
0.10 4
0.30 5
0.01
No. Σ P(x) = 0.97 13

14. Is the following a probability distribution?
EXAMPLE Identifying Probability Distributions
Is the following a probability distribution? x
P(x)
0.16 1
0.18 2
0.22 3
0.10 4
0.30 5
– 0.01
No. P(x = 5) = –0.01 14

15. Is the following a probability distribution?
EXAMPLE Identifying Probability Distributions
Is the following a probability distribution? x
P(x)
0.16 1
0.18 2
0.22 3
0.10 4
0.30 5
0.04
Yes. Σ P(x) = 1 15

16. Objective 3
Construct Probability Histograms 16

17. A probability histogram is a histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of that value of the random variable. 17

18. EXAMPLE Drawing a Probability Histogram
P(x)
0.06 1
0.58 2
0.22 3
0.10 4
0.03 5
0.01
Draw a probability histogram of the probability distribution to the right, which represents the number of movies streamed on Netflix each month. 18

19. Objective 4
Compute and Interpret the Mean of a Discrete Random Variable 19

20. The Mean of a Discrete Random Variable
The mean of a discrete random variable is given by the formula
where x is the value of the random variable and P(x) is the probability of observing the value x. 20

21. EXAMPLE Computing the Mean of a Discrete Random Variable
Compute the mean of the probability distribution to the right, which represents the number of DVDs a person rents from a video store during a single visit. x
P(x)
0.06 1
0.58 2
0.22 3
0.10 4
0.03 5
0.01 21

22. The difference between and μX gets closer to 0 as n increases.
Interpretation of the Mean of a Discrete Random Variable
Suppose an experiment is repeated n independent times and the value of the random variable X is recorded. As the number of repetitions of the experiment increases, the mean value of the n trials will approach μX, the mean of the random variable X. In other words, let x1 be the value of the random variable X after the first experiment, x2 be the value of the random variable X after the second experiment, and so on. Then
The difference between and μX gets closer to 0 as n increases. 22

23. The following data represent the number of DVDs rented by 100 randomly selected customers in a single visit. Compute the mean number of DVDs rented. 23

24. 24

25. As the number of trials of the experiment increases, the mean number of rentals approaches the mean of the probability distribution. 25

26. Objective 5
Compute and Interpret the Mean of a Discrete Random Variable 26

27. Because the mean of a random variable represents what we would expect to happen in the long run, it is also called the expected value, E(X), of the random variable. 27

28. EXAMPLE Computing the Expected Value of a Discrete Random Variable
A term life insurance policy will pay a beneficiary a certain sum of money upon the death of the policy holder. These policies have premiums that must be paid annually. Suppose a life insurance company sells a $250,000 one year term life insurance policy to a 49-year-old female for $530. According to the National Vital Statistics Report, Vol. 47, No. 28, the probability the female will survive the year is Compute the expected value of this policy to the insurance company. x
P(x)
530
530 – 250,000 = -249,470
Survives
Does not survive
E(X) = 530( ) + (-249,470)( )
= $7.50 28

29. Objective 6
Compute the Standard Deviation of a Discrete Random Variable 29

30. Standard Deviation of a Discrete Random Variable
The standard deviation of a discrete random variable X is given by
where x is the value of the random variable, μX is the mean of the random variable, and P(x) is the probability of observing a value of the random variable. 30

31. EXAMPLE. Computing the Variance and Standard Deviation
EXAMPLE Computing the Variance and Standard Deviation of a Discrete Random Variable
Compute the variance and standard deviation of the following probability distribution which represents the number of DVDs a person rents from a video store during a single visit. Remember, the mean that we found was 1.49. x
P(x)
0.06 1
0.58 2
0.22 3
0.10 4
0.03 5
0.01
6-31 31 31

32. EXAMPLE. Computing the Variance and Standard
EXAMPLE Computing the Variance and Standard Deviation of a Discrete Random Variable x μ
P(x)
1.49
–1.49
2.2201
0.06 1
–0.49
0.2401
0.58 2
0.51
0.2601
0.22 3
1.51
2.2801
0.1 4
2.51
6.3001
0.03 5
3.51
0.01
6-32 32 32

33. The Binomial Probability Distribution
Section
6.2
The Binomial Probability Distribution

34. Objectives
Determine whether a probability experiment is a binomial experiment
Compute probabilities of binomial experiments
Compute the mean and standard deviation of a binomial random variable
Construct binomial probability histograms 34

35. Criteria for a Binomial Probability Experiment
An experiment is said to be a binomial experiment if
1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial.
2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials.
3. For each trial, there are two mutually exclusive (or disjoint) outcomes, success or failure.
4. The probability of success is fixed for each trial of the experiment. 35

36. Notation Used in the Binomial Probability Distribution
There are n independent trials of the experiment.
Let p denote the probability of success so that 1 – p is the probability of failure.
Let X be a binomial random variable that denotes the number of successes in n independent trials of the experiment. So, 0 < x < n. 36

37. Which of the following are binomial experiments?
EXAMPLE Identifying Binomial Experiments
Which of the following are binomial experiments?
(a) A player rolls a pair of fair die 10 times. The number X of 7’s rolled is recorded.
Binomial experiment 37

38. Which of the following are binomial experiments?
EXAMPLE Identifying Binomial Experiments
Which of the following are binomial experiments?
(b) The 11 largest airlines had an on-time percentage of 84.7% in November, 2001 according to the Air Travel Consumer Report. In order to assess reasons for delays, an official with the FAA randomly selects flights until she finds 10 that were not on time. The number of flights X that need to be selected is recorded.
Not a binomial experiment – not a fixed number of trials. 38

39. Which of the following are binomial experiments?
EXAMPLE Identifying Binomial Experiments
Which of the following are binomial experiments?
(c) In a class of 30 students, 55% are female. The instructor randomly selects 4 students. The number X of females selected is recorded.
Not a binomial experiment – the trials are not independent. 39

40. Objective 2
Compute Probabilities of Binomial Experiments 40

41. EXAMPLE Constructing a Binomial Probability Distribution
According to the Air Travel Consumer Report, the 11 largest air carriers had an on-time percentage of 79.0% in May, Suppose that 4 flights are randomly selected from May, 2008 and the number of on-time flights X is recorded.
Construct a probability distribution for the random variable X using a tree diagram. 41

42. Binomial Probability Distribution Function
The probability of obtaining x successes in n independent trials of a binomial experiment is given by
where p is the probability of success. 42

43. “at least” or “no less than” or “greater than or equal to” ≥
Phrase Math Symbol
“at least” or “no less than” or “greater than or equal to” ≥
“more than” or “greater than” >
“fewer than” or “less than” <
“no more than” or “at most” or “less than or equal to ≤
“exactly” or “equals” or “is” = 43

44. EXAMPLE Using the Binomial Probability Distribution Function
According to the Experian Automotive, 35% of all car-owning households have three or more cars.
In a random sample of 20 car-owning households, what is the probability that exactly 5 have three or more cars? 44

45. EXAMPLE Using the Binomial Probability Distribution Function
According to the Experian Automotive, 35% of all car-owning households have three or more cars.
(b) In a random sample of 20 car-owning households, what is the probability that less than 4 have three or more cars? 45

46. EXAMPLE Using the Binomial Probability Distribution Function
According to the Experian Automotive, 35% of all car-owning households have three or more cars.
(c) In a random sample of 20 car-owning households, what is the probability that at least 4 have three or more cars? 46

47. Objective 3
Compute the Mean and Standard Deviation of a Binomial Random Variable 47

48. Mean (or Expected Value) and Standard Deviation of a Binomial Random Variable
A binomial experiment with n independent trials and probability of success p has a mean and standard deviation given by the formulas 48

49. EXAMPLE Finding the Mean and Standard Deviation of a Binomial Random Variable
According to the Experian Automotive, 35% of all car-owning households have three or more cars. In a simple random sample of 400 car-owning households, determine the mean and standard deviation number of car-owning households that will have three or more cars. 49

50. Objective 4
Construct Binomial Probability Histograms 50

51. For each histogram, comment on the shape of the distribution.
EXAMPLE Constructing Binomial Probability Histograms
(a) Construct a binomial probability histogram with n = 8 and p = 0.15.
(b) Construct a binomial probability histogram with n = 8 and p = 0. 5.
(c) Construct a binomial probability histogram with n = 8 and p = 0.85.
For each histogram, comment on the shape of the distribution. 51

52. 52

53. 53

54. 54

55. EXAMPLE Constructing Binomial Probability Histograms
Construct a binomial probability histogram with n = 25 and p = Comment on the shape of the distribution. 55

56. 56

57. EXAMPLE Constructing Binomial Probability Histograms
Construct a binomial probability histogram with n = 50 and p = Comment on the shape of the distribution. 57

58. 58

59. EXAMPLE Constructing Binomial Probability Histograms
Construct a binomial probability histogram with n = 70 and p = Comment on the shape of the distribution. 59

60. 60

61. For a fixed probability of success, p, as the number of trials n in a binomial experiment increase, the probability distribution of the random variable X becomes bell-shaped.
As a general rule of thumb, if np(1 – p) > 10, then the probability distribution will be approximately bell-shaped. 61

62. Use the Empirical Rule to identify unusual observations in a binomial experiment.
The Empirical Rule states that in a bell-shaped distribution about 95% of all observations lie within two standard deviations of the mean. 62

63. 95% of the observations lie between μ – 2σ and μ + 2σ.
Use the Empirical Rule to identify unusual observations in a binomial experiment.
95% of the observations lie between μ – 2σ and μ + 2σ.
Any observation that lies outside this interval may be considered unusual because the observation occurs less than 5% of the time. 63

64. EXAMPLE Using the Mean, Standard Deviation and Empirical Rule to Check for Unusual Results in a Binomial Experiment
According to the Experian Automotive, 35% of all car-owning households have three or more cars. A researcher believes this percentage is higher than the percentage reported by Experian Automotive. He conducts a simple random sample of 400 car-owning households and found that 162 had three or more cars. Is this result unusual ? 64

65. The result is unusual since 162 > 159.1
EXAMPLE Using the Mean, Standard Deviation and Empirical Rule to Check for Unusual Results in a Binomial Experiment
The result is unusual since 162 > 159.1 65