1. Probability Distributions
Chapter 6
Probability Distributions

2. Probability Distribution
Describes possible numerical outcomes of a chance process

3. Probability Distribution
Describes possible numerical outcomes of a chance process
Allows you to find probability of any set of possible outcomes

4. Sum of Two Dice, x Probability, p
/36
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5/36
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/36

5. Random Variables and Expected Value
Section 6.1
Random Variables and Expected Value

6. Sampling and Probability
Connection between these?

7. Sampling and Probability
Connection between these?
Sampling: Estimates a parameter (characteristic) of a population by selecting data (taking random sample) from that population

8. Sampling and Probability
Sampling: Estimates a parameter (characteristic) of a population by selecting data (taking random sample) from that population
Probability: Deals with chance of getting a specified outcome in a sample when you know about the population

9. Important Statistical Question
What might an unknown population look like?

10. Important Statistical Question
What might an unknown population look like?
Why not conduct a census to determine this?

11. Important Statistical Question
What might an unknown population look like?
Why not conduct a census to determine this?
May be impossible or impractical
Cost too much
Take too much time
Destructive data collection

12. What might an unknown population look like?
1) Use probability to learn what samples from known populations tend to look like

13. What might an unknown population look like?
1) Use probability to learn what samples from known populations tend to look like

14. What might an unknown population look like?
1) Use probability to learn what samples from known populations tend to look like
2) Take sample from unknown population & compare results to samples from known populations

15. What might an unknown population look like?
3) Populations which your sample appears to fit right in are the plausible homes for your sample

16. 500 Flips of Coin

17. Probability Distributions
Probability distributions mainly result from two different ways:
through data collection

18. Probability Distributions
Probability distributions mainly result from two different ways:
through data collection
through theory

19. Probability Distributions from Data
Suppose you are working for a contractor who is building 500 new single-family houses.
Your boss wants to know how many parking spaces will be needed per house.

20. Probability Distributions from Data
Suppose you are working for a contractor who is building 500 new single-family houses.
Your boss wants to know how many parking spaces will be needed per house.
What might happen if too few parking spaces are provided? Too many?

21. Probability Distributions from Data
Suppose you are working for a contractor who is building 500 new single-family houses.
Your boss wants to know how many parking spaces will be needed per house.
So how do you proceed?

22. Probability Distributions from Data
One method is to take a random sample of 500 households to estimate number of vehicles per household

23. Probability Distributions from Data
One method is to take a random sample of 500 households to estimate number of vehicles per household
-- takes time and money for this

24. Probability Distributions from Data
Second method is to search for existing data that you can reasonably assume represent your situation

25. Probability Distributions from Data
Second method is to search for existing data that you can reasonably assume represent your situation
Need reliable source of data
Not Wikipedia

26. Number of Motor Vehicles per Household (Display 6.1)
Number of Motor Vehicles (per household), x
Proportion of Households, P(x)
0.088 1
0.332 2
0.385 3
0.137 4
0.058
Display 6.1: The number of motor vehicles per household.
Source: U.S. Census Bureau, American Community Survey, 2004, factfinder.census.gov

27. Number of Motor Vehicles per Household (Display 6.1)
Number of Motor Vehicles (per household), x
Proportion of Households, P(x)
0.088 1
0.332 2
0.385 3
0.137 4
0.058
What about more than 4 vehicles?
Display 6.1: The number of motor vehicles per household.
Source: U.S. Census Bureau, American Community Survey, 2004, factfinder.census.gov

28. Number of Motor Vehicles per Household (Display 6.1)
Number of Motor Vehicles (per household), x
Proportion of Households, P(x)
0.088 1
0.332 2
0.385 3
0.137 4
0.058
This is actually 4 or more.
Display 6.1: The number of motor vehicles per household.
Source: U.S. Census Bureau, American Community Survey, 2004, factfinder.census.gov

29. Number of Motor Vehicles per Household (Display 6.1)
Number of Motor Vehicles (per household), x
Proportion of Households, P(x)
0.088 1
0.332 2
0.385 3
0.137 4
0.058 44
166
192.5
68.5 29
Display 6.1: The number of motor vehicles per household.
Source: U.S. Census Bureau, American Community Survey, 2004, factfinder.census.gov
500

30. Good News & Bad News
Your boss is so pleased with your work, he gives you a bonus!

31. Good News & Bad News
Your boss is so pleased with your work, he gives you a bonus!
Alas, he now knows you are a skilled statistician so you get a new project.

32. New Project
500 duplexes are also being built, so your boss wants to know something about the number of vehicles that will be parked by the two households occupying a duplex.
What do you do?

33. Way Ahead
You can take two households at random from the distribution in Display 6.1 and add their numbers.

34. Way Ahead
You can take two households at random from the distribution in Display 6.1 and add their numbers.
If you replicate this 500 times, you’ll have an approximation of the distribution your boss wants concerning duplex parking.

35. Way Ahead As typical for all modeling problems,
accuracy of result depends on
appropriateness of model used.
What assumptions do you make?

36. Assumptions
Households living in duplexes mirror households in general with respect to vehicles per household

37. Assumptions
Households living in duplexes mirror households in general with respect to vehicles per household
Number of vehicles in one household is independent of the number of vehicles in the neighboring household
How do you proceed selecting two households at random to represent a duplex?

38. Assigning Random Digits
Number of Motor Vehicles (per household), x
Proportion of Households, P(x)
0.088 1
0.332 2
0.385 3
0.137 4
0.058
Display 6.1: The number of motor vehicles per household.
Source: U.S. Census Bureau, American Community Survey, 2004, factfinder.census.gov

39. Assigning Random Digits
Number of Motor Vehicles (per household), x
Proportion of Households, P(x)
0.088 1
0.332 2
0.385 3
0.137 4
0.058
Display 6.1: The number of motor vehicles per household.
Source: U.S. Census Bureau, American Community Survey, 2004, factfinder.census.gov

40. Assigning Random Digits
Number of Motor Vehicles (per household), x
Proportion of Households, P(x)
0.088 1
0.332 2
0.385 3
0.137 4
0.058
Display 6.1: The number of motor vehicles per household.
Source: U.S. Census Bureau, American Community Survey, 2004, factfinder.census.gov

41. Assigning Random Digits
Number of Motor Vehicles (per household), x
Proportion of Households, P(x)
0.088 1
0.332 2
0.385 3
0.137 4
0.058
943 – 999, 000
Display 6.1: The number of motor vehicles per household.
Source: U.S. Census Bureau, American Community Survey, 2004, factfinder.census.gov

42. Results from one Simulation
Total Number of Motor Vehicles (per duplex), x
Proportion of Duplexes
0.008 1
0.058 2
0.142 3
0.306 4 5 6 7 8
0.250
0.160
0.064
0.010
0.002

43. Question
When you sample from distributions like the one in Display 6.1 using random digits, why should repeated random digits be used again rather than discarded?

44. A triple of random digits doesn’t represent an individual household but, along with other triples in its category, represents the many households in its category.

45. Probability Distributions from Data
You now have two ways to establish probability distributions from data:

46. Probability Distributions from Data
You now have two ways to establish probability distributions from data:
Use relative frequencies from data already collected to study a future random variable

47. Probability Distributions from Data
You now have two ways to establish probability distributions from data:
Use relative frequencies from data already collected to study a future random variable
Use simulation to build approximate probability distribution

48. Probability Distribution from Theory
Use rules of theoretical probability to build a probability distribution from basic principles and assumptions

49. Rolling Two Dice
What is the total number of possible outcomes when you roll a pair of dice?

50. Rolling Two Dice
What is the total number of possible outcomes when you roll a pair of dice?
How do you know this? 36

51. Rolling Two Dice
What is the total number of possible outcomes when you roll a pair of dice?
How do you know this?
Fundamental Principle of Counting
(multiply outcomes of each stage) 36

52. Construct Probability Distributions
Construct two probability distributions:
(1) sum of the two dice,
(2) larger number of the two dice (in case of doubles, the larger and smaller number are the same).

53. Construct Probability Distributions

54. Sum of Two Dice, x Probability, p2 3 4 5 6 7 9 10 11 12

55. Sum of Two Dice, x Probability, p
/36
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5/36
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56. Larger Number Probability Distribution
Larger Number, x Probability, p

57. Larger Number
Larger Number, x Probability, p 1 2 3 4 5 6

58. Larger Number
Larger Number, x Probability, p
/36 2 3 4 5 6

59. Construct Probability Distributions

60. Larger Number Larger Number, x Probability, p 1 1/36 2 3/36 3 5/36
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Total

61. Sampling Lung Cancer Patients
Suppose two lung cancer patients are randomly selected from the large population of people with that disease.
Construct the probability distribution of X, the number of patients with lung cancer caused by smoking.

62. Sampling Lung Cancer Patients
Suppose two lung cancer patients are randomly selected from the large population of people with that disease. Construct the probability distribution of X, the number of patients with lung cancer caused by smoking.
Lung Cancer Cases Proportion
Smoking responsible 0.9
Smoking not responsible 0.1

63. Sampling Lung Cancer Patients
Lung Cancer Cases Proportion
Smoking responsible 0.9
Smoking not responsible 0.1
How many possible outcomes are there for the two patients?

64. Sampling Lung Cancer Patients
Lung Cancer Cases Proportion
Smoking responsible 0.9
Smoking not responsible 0.1
How many possible outcomes are there for the two patients?
2●2 = 4

65. Sampling Lung Cancer Patients
Lung Cancer Cases Proportion
Smoking responsible 0.9
Smoking not responsible 0.1
What are the 4 possible
outcomes for the two patients?

66. Sampling Lung Cancer Patients
Lung Cancer Cases Proportion
Smoking responsible 0.9
Smoking not responsible 0.1
What are the possible outcomes for the two patients? 4 possible outcomes
No for 1st patient and no for 2nd patient
No for 1st patient and yes for 2nd patient
Yes for 1st patient and no for 2nd patient
Yes for 1st patient and yes for 2nd patient

67. Sampling Lung Cancer Patients
Lung Cancer Cases Proportion
Smoking responsible 0.9
Smoking not responsible 0.1
P(No for 1st patient and no for 2nd patient) = P(no for 1st) P( no for 2nd) = (0.1)(0.1) = 0.01
No for 1st patient and yes for 2nd patient
Yes for 1st patient and no for 2nd patient
Yes for 1st patient and yes for 2nd patient

68. Sampling Lung Cancer Patients
Lung Cancer Cases Proportion
Smoking responsible 0.9
Smoking not responsible 0.1
P(No for 1st patient and no for 2nd patient) = P(no for 1st) P( no for 2nd) = (0.1)(0.1) = 0.01
P(No,yes) = (0.1)(0.9) = 0.09
P(Yes,no) = (0.9)(0.1) = 0.09
P(Yes,yes) = (0.9)(0.9) = 0.81

69. Sampling Lung Cancer Patients
Probability distribution of X, the number of patients with lung cancer, is:
x Probability of x 1 2

70. Sampling Lung Cancer Patients
Probability distribution of X, the number of patients with lung cancer, is:
x Probability of x
= 0.18

71. Sampling Lung Cancer Patients
Probability distribution of X, the number of patients with lung cancer, is:
x Probability of x

72. Questions?