1. Discrete Random Variables
Statistics
Discrete Random Variables

2. Random Variables
A random variable

3. Random Variables
A random variable “varies” (not always the same)

4. Random Variables
A random variable “varies” is “random”

5. Random Variables Random variable… hmmm… what’s that?

6. Random Variables
A random variable… Is a way to quantify outcomes of unforecastable processes

7. Random Variables
For a coin toss: X is a random variable that assigns a number to an outcome

8. Random Variables
This allows us to do arithmetic with the outcomes

9. Random Variables
Fred + Angela – Juan = ? Not easy!

10. Random Variables
1 + 7 – 4.2 = ? Much easier!

11. Random Variables
There are two types of random variables:

12. Random Variables
There are two types of random variables: discrete

13. Random Variables
There are two types of random variables: discrete continuous

14. Random Variables
A discrete variable has countable values, such as a list of non-negative integers

15. Random Variables
Or the list of people in a club

16. Random Variables
The values are distinct or separate

17. Random Variables
Not discreet (which means on the down low, under the radar)

18. Random Variables
There are two types of random variables: discrete continuous

19. Random Variables
A continuous variable can take on any value in an interval

20. Which? TYPES OF STATISTICS IN-CLASS PROBLEM

21. Which? TYPES OF STATISTICS IN-CLASS PROBLEM

22. Random Variables
A discrete variable can have an infinite number of outcomes

23. Random Variables
A discrete variable can have an infinite number of outcomes “countably infinite”

24. Which? TYPES OF STATISTICS IN-CLASS PROBLEM

25. Which? TYPES OF STATISTICS IN-CLASS PROBLEM

26. Questions?

27. Discrete Probability
P(X=x) P(x) means “the probability that the random variable X equals the value x”

28. Discrete Probability
Remember “Σ” means “the sum of”

29. Discrete Probability
Rules for discrete probabilities: Σ P(x) = 1 or 100%

30. Discrete Probability
Rules for discrete probabilities: Σ P(x) = 1 or 100% 0 ≤ P(x) ≤ 1 or 100%

31. Discrete Probability
A probability histogram:

32. Discrete Probability
A lot of variables can have only two values: M/F H/T Black/White On/Off 1/0

33. Binomial Probability
Variables that can have only two values are called: “binomial” The values are mutually exclusive events

34. Binomial Probability
The probability of one of the values occurring is called “p” The probability of the other value occurring is called “q”

35. Binomial Probability
p + q = 1 or 100%

36. Binomial Probability
A binomial experiment:

37. Binomial Probability
A binomial experiment: is performed a fixed number of times

38. Binomial Probability
A binomial experiment: is performed a fixed number of times each repetition is called a “trial”

39. Binomial Probability
A binomial experiment: the trials are independent

40. Binomial Probability
A binomial experiment: the trials are independent the outcome of one trial will not affect the outcome of another trial

41. Binomial Probability
A binomial experiment: for each trial, there are two mutually exclusive outcomes: success or failure

42. Binomial Probability
A binomial experiment: the probability of success is the same for each trial

43. Binomial Probability
Notation: “n” trials

44. Binomial Probability
Notation: “n” trials “p” is the probability of success

45. Binomial Probability
Notation: “n” trials “p” is the probability of success “q” or “1-p” is the probability of failure

46. Binomial Probability
Notation: “n” trials “p” is the probability of success “q” or “1-p” is the probability of failure “X” is the number of successes in the “n” trials

47. Binomial Probability
0 ≤ p ≤ 1 0 ≤ q ≤ 1 and: 0 ≤ x ≤ n

48. Binomial Probability
Binomial Experiment Rules:

49. Binomial Probability
Binomial Experiment Rules: You must have a fixed number of trials

50. Binomial Probability
Binomial Experiment Rules: You must have a fixed number of trials Each trial is an independent event

51. Binomial Probability
Binomial Experiment Rules: You must have a fixed number of trials Each trial is an independent event There are only two outcomes

52. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
Binomial or not? Tossing a coin a hundred times to see how many land on heads

53. Binomial or not? Tossing a coin until you get heads
BINOMIAL PROBABILITY
IN-CLASS PROBLEM
Binomial or not? Tossing a coin until you get heads

54. Binomial or not? Asking 100 people how much they weigh
BINOMIAL PROBABILITY
IN-CLASS PROBLEM
Binomial or not? Asking 100 people how much they weigh

55. Binomial or not? Asking 100 people if they have ever been to Paris
BINOMIAL PROBABILITY
IN-CLASS PROBLEM
Binomial or not? Asking 100 people if they have ever been to Paris

56. Questions?

57. Binomial Probability
Remember nCx is the number of ways of obtaining x successes in n trials

58. Binomial Probability
The probability of obtaining x successes in n independent trials of a binomial experiment: P(x) = nCx px(1-p)n-x or: P(x) = nCx px(q)n-x

59. Binomial Probability
To work a binomial problem:

60. Binomial Probability
To work a binomial problem: What is a “Success”? Success must be for a single trial

61. Binomial Probability
To work a binomial problem: What is the probability of success “p”?

62. Binomial Probability
To work a binomial problem: What is the probability of failure “q”?

63. Binomial Probability
To work a binomial problem: What is the number of trials?

64. Binomial Probability
To work a binomial problem: What is the number of successes out of those trials needed?

65. Binomial Probability To work a binomial problem: What is a “Success”?
What is the probability of success “p”?
What is the probability of failure “q”?
What is the number of trials?
What is the number of successes out
of those trials needed?

66. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die?

67. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? What is a “Success”?

68. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? What is a “Success”? Success = "Rolling a 6 on a single die"

69. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? What is the probability of success?

70. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? What is the probability of success? p = 1/6

71. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? What is the probability of failure?

72. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? What is the probability of failure? q = 5/6

73. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? What is the number of trials?

74. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? What is the number of trials? n = 6

75. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? What is the number of successes out of those trials needed?

76. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? What is the number of successes out of those trials needed? x = 2

77. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? You could list the outcomes: FFFFFS FFFFSS FFFSSS FFSSSS FSSSSS SSSSSS FFFSFS FFSFFS FSFFFS SFFFFS SFFFSS SFFSSS SFSSSS SFSFFS SSFFFS …
Aagh!!!

78. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
Remember: The probability of getting exactly x success in n trials, with the probability of success on a single trial being p is: P(x) = nCx × px × qn-x

79. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? P(2) = 6C2 × (1/6)2 × (5/6)6-2

80. BINOMIAL PROBABILITY
IN-CLASS PROBLEM
What is the probability of rolling exactly two sixes in 6 rolls of a die? P(2) = 6C2 × (1/6)2 × (5/6)6-2 = 15 × .028 × .48 ≈ .20

81. Questions?

82. Binomial Probability
The mean and standard deviation of a binomial are easy!

83. Binomial Probability
The mean of a binomial experiment: μx = np

84. Binomial Probability
The variance of a binomial experiment: σx2 = np(1−p) or: σx2 = npq

85. Binomial Probability
The standard deviation of a binomial experiment: σx = np(1−p) or: σx = npq

86. Binomial Probability A binomial distribution histogram:

87. TYPES OF STATISTICS
IN-CLASS PROBLEM
What is p?

88. TYPES OF STATISTICS
IN-CLASS PROBLEM
What is p? p = 0.2

89. TYPES OF STATISTICS
IN-CLASS PROBLEM
What is q?

90. TYPES OF STATISTICS
IN-CLASS PROBLEM
What is q? q = 1-p = 1-.2 = .8

91. TYPES OF STATISTICS
IN-CLASS PROBLEM
What is n?

92. TYPES OF STATISTICS
IN-CLASS PROBLEM
What is n? n = 15

93. TYPES OF STATISTICS
IN-CLASS PROBLEM
What is μx?

94. TYPES OF STATISTICS
IN-CLASS PROBLEM
What is μx? μx = np = 15×.2 = 3

95. TYPES OF STATISTICS
IN-CLASS PROBLEM
What is σx2?

96. What is σx2? σx2 = npq = 15×.2×.8 = 2.4 TYPES OF STATISTICS
IN-CLASS PROBLEM
What is σx2? σx2 = npq = 15×.2×.8 = 2.4

97. TYPES OF STATISTICS
IN-CLASS PROBLEM
What is σx?

98. What is σx? σx = npq = 15×.2×.8 ≈ 1.5 TYPES OF STATISTICS
IN-CLASS PROBLEM
What is σx? σx = npq = 15×.2×.8 ≈ 1.5

99. Questions?

100. Poisson Probability
The Poisson distribution was introduced by the French mathematician Siméon Denis Poisson

101. Poisson Probability
It is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event

102. Poisson Probability
For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day

103. Poisson Probability
If receiving any particular piece of mail does not affect the arrival times of future pieces of mail, the number of pieces of mail received in a day exhibits a Poisson distribution

104. Poisson Probability
The number of students who arrive at the student union per minute will likely not follow a Poisson distribution: the rate is not constant (low rate during class time, high rate between class times)

105. Poisson Probability
The number of students who arrive at the student union per minute will likely not follow a Poisson distribution: the arrivals of individual students are not independent (students tend to come in groups)

106. Poisson Probability
The number of magnitude 5 earthquakes per year in a country would not follow a Poisson distribution because one large earthquake increases the probability of aftershocks of similar magnitude

107. Poisson Probability
Among patients admitted to the intensive care unit of a hospital, the number of days that the patients spend in the ICU is not Poisson distributed because the number of days cannot be zero

108. Poisson Probability
The Poisson distribution may be useful to model events such as: -The number of meteorites greater than 1 meter diameter that strike Earth in a year -The number of patients arriving in an emergency room between 10 and 11 pm

109. Poisson Probability
The Poisson distribution is an appropriate if: -k is the number of times an event occurs in an interval -events occur independently -the rate at which events occur is constant

110. Poisson Probability
The Poisson distribution is an appropriate if: -two events cannot occur at exactly the same instant -the probability of an event in a small sub-interval is proportional to the length of the sub-interval

111. Poisson Probability
The Poisson distribution is an appropriate if: the actual probability distribution is given by a binomial distribution and the number of trials is sufficiently bigger than the number of successes one is asking about

112. Poisson Probability
The average number of events in an interval is designated λ (lambda) The probability of observing k events in an interval is given by: P = e −λ λ 𝒌 k!

113. Poisson Probability
As you can imagine, it is painful to calculate

114. Poisson Probability
Excel has a formula:

115. Poisson Probability Excel has a formula: λ k False
(for an individual probability)
True
(for a cumulative probability)

116. Poisson Probability
Or:

117. Hypergeometric
The hypergeometric distribution describes the probability of k successes in n draws, without replacement, from a finite population of size N that contains exactly K objects with that feature, wherein each draw is either a success or a failure

118. Hypergeometric
In contrast, the binomial distribution describes the probability of k successes in n draws with replacement

119. Hypergeometric
In Excel:

120. Hypergeometric
Or use:

121. Hypergeometric
The test is often used to identify which sub-populations are over- or under-represented in a sample

122. Hypergeometric
For example, a marketing group could use the test to understand their customer base by testing a set of known customers for over-representation of people under 30

123. Questions?