1. Welcome to .
Week 07 Tues .
MAT135 Statistics

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 PROBLEMs 1,2

21. Which? TYPES OF STATISTICS IN-CLASS PROBLEM 3

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 PROBLEMS 4,5

25. Which? TYPES OF STATISTICS IN-CLASS PROBLEM 6

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 7
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 8
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 9
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 10
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 11
What is the probability of rolling exactly two sixes in 6 rolls of a die?

67. BINOMIAL PROBABILITY
IN-CLASS PROBLEM 11
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 11
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 11
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 11
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 11
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 11
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 11
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 11
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 11
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 11
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 11
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 11
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 11
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 11
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

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 15
What is p?

88. TYPES OF STATISTICS
IN-CLASS PROBLEM 15
What is p? p = 0.2

89. TYPES OF STATISTICS
IN-CLASS PROBLEM 15
What is q?

90. TYPES OF STATISTICS
IN-CLASS PROBLEM 15
What is q? q = 1-p = 1-.2 = .8

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

92. TYPES OF STATISTICS
IN-CLASS PROBLEM 12
What is n? n = 15

93. TYPES OF STATISTICS
IN-CLASS PROBLEM 12
What is μx?

94. What is μx? μx = np = 15×.2 = 3 TYPES OF STATISTICS
IN-CLASS PROBLEM 12
What is μx? μx = np = 15×.2 = 3

95. TYPES OF STATISTICS
IN-CLASS PROBLEM 12
What is σx2?

96. TYPES OF STATISTICS
IN-CLASS PROBLEM 12
What is σx2?

97. What is σx2? σx2 = npq = 15×.2×.8 = 2.4 TYPES OF STATISTICS
IN-CLASS PROBLEM 12
What is σx2? σx2 = npq = 15×.2×.8 = 2.4

98. TYPES OF STATISTICS
IN-CLASS PROBLEM 12
What is σx?

99. What is σx? σx = npq = 15×.2×.8 ≈ 1.5 TYPES OF STATISTICS
IN-CLASS PROBLEM 12
What is σx? σx = npq = 15×.2×.8 ≈ 1.5

100. Questions?

101. You survived!
Turn in your homework! Don’t forget your homework due next class! See you Thursday!