1. Last Time Histograms Notions of Center
Binomial Probability Distributions
Lists of Numbers
Real Data
Excel Computation
Notions of Center
Average of list of numbers
Weighted Average

2. Administrative Matters
Midterm I, coming Tuesday, Feb. 24
Excel notation to avoid actual calculation
So no computers or calculators
Bring sheet of formulas, etc.
No blue books needed
(will just write on my printed version)

3. Administrative Matters
Midterm I, coming Tuesday, Feb. 24
Material Covered:
HW 1 – HW 5
Note: due Thursday, Feb. 19
Will ask grader to return Mon. Feb. 23
Can pickup in my office (Hanes 352)
So this weeks HW not included

4. Administrative Matters
Midterm I, coming Tuesday, Feb. 24
Extra Office Hours:
Monday, Feb :00 – 9:00
Monday, Feb :00 – 10:00
Monday, Feb :00 – 11:00
Tuesday, Feb :00 – 9:00
Tuesday, Feb :00 – 10:00
Tuesday, Feb :00 – 2:00

5. Administrative Matters
Midterm I, coming Tuesday, Feb. 24
How to study:
Rework HW problems
Since problems come from there
Actually do, not “just look over”
In random order (as on exam)
Print HW sheets, use as a checklist
Work Practice Exam
Posted in Blackboard “Course Information” Area

6. Reading In Textbook Approximate Reading for Today’s Material:
Pages ,
Approximate Reading for Next Class:
Pages 55-68,

7. Big Picture Margin of Error Choose Sample Size Need better prob tools
Start with visualizing probability distributions

8. Big Picture Margin of Error Choose Sample Size Need better prob tools
Start with visualizing probability distributions,
Next exploit constant shape property of Bi

9. Big Picture Start with visualizing probability distributions,
Next exploit constant shape property of Binom’l

10. Big Picture Start with visualizing probability distributions,
Next exploit constant shape property of Binom’l
Centerpoint feels p

11. Big Picture Start with visualizing probability distributions,
Next exploit constant shape property of Binom’l
Centerpoint feels p Spread feels n

12. Big Picture Start with visualizing probability distributions,
Next exploit constant shape property of Binom’l
Centerpoint feels p Spread feels n
Now quantify these ideas, to put them to work

13. Will later study “notions of spread”
Notions of Center
Will later study “notions of spread”

14. Notions of Center Textbook: Sections 4.4 and 1.2
Recall parallel development:
(a) Probability Distributions
(b) Lists of Numbers
Study 1st, since easier

15. Notions of Center Lists of Numbers
“Average” or “Mean” of x1, x2, …, xn
Mean = =
common notation

16. Notions of Center Generalization of Mean: “Weighted Average”
Intuition: Corresponds to finding balance point of weights on number line

17. Notions of Center Generalization of Mean: “Weighted Average”
Intuition: Corresponds to finding balance point of weights on number line

18. Notions of Center Textbook: Sections 4.4 and 1.2
Recall parallel development:
(a) Probability Distributions
(b) Lists of Numbers

19. Notions of Center Probability distributions, f(x)
Approach: use connection to lists of numbers

20. Notions of Center Probability distributions, f(x)
Approach: use connection to lists of numbers
Recall: think about many repeated draws

21. Notions of Center Probability distributions, f(x)
Approach: use connection to lists of numbers
Draw X1, X2, …, Xn from f(x)

22. Notions of Center Probability distributions, f(x)
Approach: use connection to lists of numbers
Draw X1, X2, …, Xn from f(x)
Compute and express in terms of f(x)

23. Notions of Center

24. Notions of Center
Rearrange list, depending on values

25. Notions of Center
Number of Xis that are 1

26. Notions of Center
Apply Distributive Law of Arithmetic

27. Notions of Center
Recall “Empirical Probability Function”

28. Notions of Center

29. Notions of Center
Frequentist approximation

30. Notions of Center

31. Notions of Center
A weighted average of values that X takes on

32. Notions of Center
A weighted average of values that X takes on, where weights are probabilities

33. Notions of Center This concept deserves its own name: Expected Value
A weighted average of values that X takes on, where weights are probabilities
This concept deserves its own name:
Expected Value

34. Expected Value
Define Expected Value of a random variable X:

35. Expected Value
Define Expected Value of a random variable X:

36. Expected Value Define Expected Value of a random variable X:
Useful shorthand notation

37. Expected Value Define Expected Value of a random variable X:
Recall f(x) = 0, for most x, so sum
only operates for values X takes on

38. Expected Value
E.g. Roll a die, bet (as before):

39. Expected Value E.g. Roll a die, bet (as before):
Win $9 if 5 or 6, Pay $4, if 1, 2 or 3, otherwise (4) break even

40. Expected Value E.g. Roll a die, bet (as before):
Win $9 if 5 or 6, Pay $4, if 1, 2 or 3, otherwise (4) break even
Let X = “net winnings”

41. Expected Value E.g. Roll a die, bet (as before):
Win $9 if 5 or 6, Pay $4, if 1, 2 or 3, otherwise (4) break even
Let X = “net winnings”

42. Expected Value E.g. Roll a die, bet (as before):
Win $9 if 5 or 6, Pay $4, if 1, 2 or 3, otherwise (4) break even
Let X = “net winnings”
Are you keen to play?

43. Expected Value
Let X = “net winnings”

44. Expected Value
Let X = “net winnings”
Weighted average, wts & values

45. Expected Value
Let X = “net winnings”
Weighted average, wts & values

46. Expected Value Let X = “net winnings”
i.e. weight average of values 9, -4 & 0, with
weights of “how often expect”, thus “expected”

47. Expected Value Let X = “net winnings”
Conclusion: on average in many plays, expect
to win $1 per play.

48. Expected Value
Caution: “Expected value” is not what is expected on one play
(which is either 9, -4 or 0)
But instead on average, over many plays
HW: , (1.9, 1)

49. Expected Value Real life applications of expected value:
Decision Theory
Operations Research
Rational basis for making business decisions
In presence of uncertainty
Common Goal: maximize expected profits
Gives good average results over long run

50. (+ from their perspective)
Expected Value
Real life applications of expected value:
Decision Theory
Casino Gambling
Casino offers games with + expected value
(+ from their perspective)
Their goal: good overall average performance
Expected Value is a useful tool for this

51. Expected Value Real life applications of expected value:
Decision Theory
Casino Gambling
Insurance
Companies make profit
By writing policies with + expected value
Their goal is long run average performance

52. Expected Value Real life applications of expected value:
Decision Theory
Casino Gambling
Insurance
State Lotteries
State’s view: games with + expected value
Raise money for state in long run overall

53. Flip Side of Expected Value
Decisions made against expected value:

54. Flip Side of Expected Value
Decisions made against expected value:
Casino Gambling
Why do people play?

55. Flip Side of Expected Value
Decisions made against expected value:
Casino Gambling
Why do people play?
Odds are against them
For sure will lose “over long run”

56. Flip Side of Expected Value
Decisions made against expected value:
Casino Gambling
Why do people play?
Odds are against them
For sure will lose “over long run”
But love of short run successes, can make eventual long term loss worthwhile
Are buying entertainment

57. Flip Side of Expected Value
Decisions made against expected value:
Casino Gambling
Insurance
Why should you buy it?

58. Flip Side of Expected Value
Decisions made against expected value:
Casino Gambling
Insurance
Why should you buy it?
You lose in expected value sense

59. Flip Side of Expected Value
Decisions made against expected value:
Casino Gambling
Insurance
Why should you buy it?
You lose in expected value sense
But E not applicable since you only play once
Avoids chance of catastrophic loss
Allows low cost sharing of risk

60. Flip Side of Expected Value
Decisions made against expected value:
Casino Gambling
Insurance
State Lotteries
Why do people play?

61. Flip Side of Expected Value
Decisions made against expected value:
Casino Gambling
Insurance
State Lotteries
Why do people play?
Clear loss in expected value

62. Flip Side of Expected Value
Decisions made against expected value:
Casino Gambling
Insurance
State Lotteries
Why do people play?
Clear loss in expected value
But only play once (hopefull), so not applicable
Worth feeling of hope from buying ticket?

63. Flip Side of Expected Value
Interesting issues about State Lotteries:
A very different type of tax
Big Plus:
Only totally voluntary tax
Big Minus:
Tax paid mostly by poor

64. Flip Side of Expected Value
Decisions made against expected value:
Key Lesson: Expected Value tells what happens on average over long run, not in one play

65. Flip Side of Expected Value
Decisions made against expected value:
Key Lesson: Expected Value tells what happens on average over long run, not in one play
Conclude Expected Value not good for everything, but very good for many things

66. Another Inverse View of Expected Value
Suppose you have $5000, and need $10,000

67. Another Inverse View of Expected Value
Suppose you have $5000, and need $10,000
e.g. you owe mafia $5000 (gambling debt?), clean out safe at work for $5000.

68. Another Inverse View of Expected Value
Suppose you have $5000, and need $10,000
e.g. you owe mafia $5000 (gambling debt?), clean out safe at work for $5000.
If give to mafia, you go to jail

69. Another Inverse View of Expected Value
Suppose you have $5000, and need $10,000
e.g. you owe mafia $5000 (gambling debt?), clean out safe at work for $5000.
If give to mafia, you go to jail, so decide to raise another $5000 by gambling.

70. Another Inverse View of Expected Value
Suppose you have $5000, and need $10,000 and can make even bets, with P[win] = 0.48

71. Another Inverse View of Expected Value
Suppose you have $5000, and need $10,000 and can make even bets, with P[win] = 0.48
Can really do this, e.g. bet on Red in game of Roulette at a casino

72. Another Inverse View of Expected Value
Suppose you have $5000, and need $10,000 and can make even bets, with P[win] = 0.48
Important question:
Make one large bet? Or many small bets?
Something in between?

73. Another Inverse View of Expected Value
Suppose you have $5000, and need $10,000 and can make even bets, with P[win] = 0.48
Make one large bet? Or many small bets?
E[Gain] = 0 x P[loss] + $2 x P[win]
= 0 x (0.52) + $2 x (0.48) = $0.96

74. Another Inverse View of Expected Value
Suppose you have $5000, and need $10,000 and can make even bets, with P[win] = 0.48
Make one large bet? Or many small bets?
E[Gain] = $0.96
Interpretation: “expect” to lose $0.04 every time you play

75. Another Inverse View of Expected Value
Suppose you have $5000, and need $10,000 and can make even bets, with P[win] = 0.48
Make one large bet? Or many small bets?
E[Gain] = $0.96
Interpretation: “expect” to lose $0.04 every time you play
Why games are so profitable for casinos

76. Another Inverse View of Expected Value
Suppose you have $5000, and need $10,000 and can make even bets, with P[win] = 0.48
Make one large bet? Or many small bets?
E[Gain] = $0.96
Interpretation: “expect” to lose $0.04 every time you play
Many plays  Expected Value dictates result

77. Another Inverse View of Expected Value
Suppose you have $5000, and need $10,000 and can make even bets, with P[win] = 0.48
Make one large bet? Or many small bets?
E[Gain] = $0.96
Interpretation: “expect” to lose $0.04 every time you play
So best to make just one large bet!

78. Another Inverse View of Expected Value
Interpretation: “expect” to lose $0.04 every time you play
So best to make just one large bet!

79. Another Inverse View of Expected Value
Interpretation: “expect” to lose $0.04 every time you play
So best to make just one large bet!
After many plays, will surely lose!
(lesson of expected value)

80. Another Inverse View of Expected Value
Another View:
Strategy P[win $1,0000]
one $5000 bet ≈ ½

81. Another Inverse View of Expected Value
Another View:
Strategy P[win $1,0000]
one $5000 bet ≈ ½
two $2500 bets ≈ (0.48)2 ≈ ¼

82. Another Inverse View of Expected Value
Another View:
Strategy P[win $1,0000]
one $5000 bet ≈ ½
two $2500 bets ≈ (0.48)2 ≈ ¼
four $1250 bets ≈ 1/16

83. Another Inverse View of Expected Value
Another View:
Strategy P[win $1,0000]
one $5000 bet ≈ ½
two $2500 bets ≈ (0.48)2 ≈ ¼
four $1250 bets ≈ 1/16
many bets no chance

84. Another Inverse View of Expected Value
Interpretation: “expect” to lose $0.04 every time you play
So best to make just one large bet!
Casino Folklore: Sometimes people really make such bets

85. Expected Value
Binomial Expected Value:
For X ~ Bi(n,p),

86. Expected Value Binomial Expected Value:
For X ~ Bi(n,p), Expected Value is
probability weighted average of values

87. Expected Value Binomial Expected Value:
For X ~ Bi(n,p), Expected Value is
Use Binomial Probability Distribution

88. Expected Value Binomial Expected Value:
For X ~ Bi(n,p), Expected Value is
After a long and tricky calculation
(details beyond scope of this course)

89. “Expect” to win proportion p, of n trials
Expected Value
For X ~ Bi(n,p),
Makes sense:
“Expect” to win proportion p, of n trials

90. “Expect” to win proportion p, of n trials
Expected Value
For X ~ Bi(n,p),
Makes sense:
“Expect” to win proportion p, of n trials
Just use this formula from here on out

91. “Expect” to win proportion p, of n trials
Expected Value
For X ~ Bi(n,p),
Makes sense:
“Expect” to win proportion p, of n trials
Just use this formula from here on out
E.g. to capture “shifting mean”

92. Expected Value For X ~ Bi(n,p), HW: 5.28a, mean part only (900)

93. Properties of Expected Value
Linearity:
For X ~ f(x) and Y ~ g(y)
E(aX + bY) =

94. Properties of Expected Value
Linearity:
For X ~ f(x) and Y ~ g(y)
E(aX + bY) = Σx Σy (ax + by) f(x) g(y)
Weighted average, where weights
are probabilities

95. Properties of Expected Value
Linearity:
For X ~ f(x) and Y ~ g(y)
E(aX + bY) = Σx Σy (ax + by) f(x) g(y) =
= Σx Σy (ax f(x) g(y) + by f(x) g(y))
(Distributive Rule)

96. Properties of Expected Value
Linearity:
For X ~ f(x) and Y ~ g(y)
E(aX + bY) = Σx Σy (ax + by) f(x) g(y) =
= Σx Σy (ax f(x) g(y) + by f(x) g(y)) =
= Σx Σy axf(x)g(y) + Σx Σy axf(x)g(y)
(Associative Property of Addition)

97. Properties of Expected Value
Linearity:
For X ~ f(x) and Y ~ g(y)
E(aX + bY) = Σx Σy (ax + by) f(x) g(y) =
= Σx Σy (ax f(x) g(y) + by f(x) g(y)) =
= Σx Σy axf(x)g(y) + Σx Σy axf(x)g(y) =
= (Σxaxf(x)) Σyg(y) + (Σxf(x)) Σybyg(y)
(Distributive Rule)

98. Properties of Expected Value
Linearity: For X ~ f(x) and Y ~ g(y)
E(aX + bY) =
= (Σxaxf(x)) Σyg(y) + (Σxf(x)) Σybyg(y)

99. Properties of Expected Value
Linearity: For X ~ f(x) and Y ~ g(y)
E(aX + bY) =
= (Σxaxf(x)) Σyg(y) + (Σxf(x)) Σybyg(y) =
= Σx a x f(x) + Σy b y g(y)
(Since Σx f(x) = Σy y g(y) = 1)

100. Properties of Expected Value
Linearity: For X ~ f(x) and Y ~ g(y)
E(aX + bY) =
= (Σxaxf(x)) Σyg(y) + (Σxf(x)) Σybyg(y) =
= Σx a x f(x) + Σy b y g(y) =
= a Σx x f(x) + b Σy y g(y)
(Distributive Rule)

101. Properties of Expected Value
Linearity: For X ~ f(x) and Y ~ g(y)
E(aX + bY) =
= (Σxaxf(x)) Σyg(y) + (Σxf(x)) Σybyg(y) =
= Σx a x f(x) + Σy b y g(y) =
= a Σx x f(x) + b Σy y g(y) =
= a E(X) + b E(Y)

102. Properties of Expected Value
Linearity: For X ~ f(x) and Y ~ g(y)
E(aX + bY) = a E(X) + b E(Y)
i.e. E(linear combo) = linear combo (E)

103. Properties of Expected Value
HW:
(mean part only)
(mean part only)

104. Properties of Expected Value
HW: C18 An insurance company sells 1378 policies to cover bicycles against theft for 1 year. It costs $300 to replace a stolen bicycle and the probability of theft is estimated at Suppose there is no chance of more than one theft per individual.

105. Properties of Expected Value
HW: C18 (cont.)
Calculate the expected payout for each policy, to give a break even price for each policy. ($24)
If 2 times the break even price is actually charged, what is the company’s expected profit per policy, if the theft rate is actually 0.10? ($18)

106. Research Corner
Recall Hidalgo
StampData &
Movie over binwidth

107. Research Corner Recall Hidalgo StampData & Movie over binwidth
Main point:
Binwidth drives
histogram performance

108. Research Corner Less known fact: Bin location also has Serious effect
(even for fixed width)

109. Research Corner
How many bumps?
~2?

110. Research Corner
How many bumps?
~3?

111. Research Corner
How many bumps?
~7?

112. Research Corner
How many bumps?
~7?

113. Research Corner Explanation? Compare with “smoothed version” called
“Kernel Density Estimate”

114. Research Corner Compare with “smoothed version” called
“Kernel Density Estimate”
Peaks appear:
when entirely in a bin

115. Research Corner Compare with “smoothed version” called
“Kernel Density Estimate”
Peaks disappear:
when split between
two bins bin

116. Research Corner
Question: If understand problem with histogram, using Kernel Density Estimate
Then why not use KDE for data analysis?
Will explore KDE later.

117. Works well for ~symmetric distributions
Notions of Center
Caution about mean:
Works well for ~symmetric distributions

118. Works well for ~symmetric distributions
Notions of Center
Caution about mean:
Works well for ~symmetric distributions
E.g. Buffalo Snowfalls

119. Works well for ~symmetric distributions
Notions of Center
Caution about mean:
Works well for ~symmetric distributions
E.g. Buffalo Snowfalls
Analyzed in:

120. Works well for ~symmetric distributions
Notions of Center
Caution about mean:
Works well for ~symmetric distributions
E.g. Buffalo Snowfalls
Mean = 80.3
(from Excel)

121. Works well for ~symmetric distributions
Notions of Center
Caution about mean:
Works well for ~symmetric distributions
E.g. Buffalo Snowfalls
Mean = 80.3
(from Excel)

122. Works well for ~symmetric distributions
Notions of Center
Caution about mean:
Works well for ~symmetric distributions
E.g. Buffalo Snowfalls
Mean = 80.3
Visually sensible
Notion of “Center”

123. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions

124. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data

125. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
Time (in days) to suicide attempt
Of Suicide Patients
After Initial Treatment

126. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
Analyzed in:

127. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data

128. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
Clearly not mound shaped

129. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
Clearly not mound shaped
Very asymmetric

130. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
Clearly not mound shaped
Very asymmetric
Called “right skewed”

131. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
Mean = 122.3

132. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
Mean = 122.3

133. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
Mean = 122.3
Sensible as “center”??

134. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
Mean = 122.3
Sensible as “center”??
%(data ≥) = 30.2%

135. But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
Mean = 122.3
Sensible as “center”??
%(data ≥) = 30.2%
Too Small…

136. Notions of Center Perhaps better notion of “center”:
Take center to be point in middle
I.e. have 50% of data smaller
And 50% of data larger
This is called the “median”

137. Notions of Center
Median: = Value in middle (of sorted list)

138. Notions of Center Median: = Value in middle (of sorted list)
Unsorted E.g: Sorted E.g:

139. Notions of Center Median: = Value in middle (of sorted list)
Unsorted E.g: Sorted E.g:
One in middle???

140. Notions of Center Median: = Value in middle (of sorted list)
Unsorted E.g: Sorted E.g:
One in middle??? NO, must sort

141. Notions of Center Median: = Value in middle (of sorted list)
Unsorted E.g: Sorted E.g:
Sensible version of “middle”

142. Notions of Center
What about ties?
Sorted E.g: 1 2 3

143. Notions of Center What about ties? Sorted E.g: Tie for point in 1
Tie for point in 1
middle 3

144. Notions of Center What about ties? Sorted E.g: Tie for point in 1
Tie for point in 1
middle 3
Break by taking average (of two tied values):

145. Notions of Center What about ties? Sorted E.g: Tie for point in 1
Tie for point in 1
middle 3
Break by taking average (of two tied values):
e.g. Median = 1.5

146. Notions of Center Median: = Value in middle (of sorted list)
Unsorted E.g: Sorted E.g:
EXCEL: use function “MEDIAN”

147. Notions of Center EXCEL: use function “MEDIAN”
Very similar to other functions
E.g. see:

148. Notions of Center
E.g. Buffalo Snowfalls
Mean = 80.3

149. Notions of Center E.g. Buffalo Snowfalls Mean = 80.3 Median = 79.6
(from Excel)

150. Notions of Center E.g. Buffalo Snowfalls Mean = 80.3 Median = 79.6
Very similar

151. Notions of Center E.g. Buffalo Snowfalls Mean = 80.3 Median = 79.6
Very similar
(expected from
symmetry)

152. Notions of Center E.g. British Suicides Data Mean = 122.3
Median = 77.5
Substantially different

153. Notions of Center E.g. British Suicides Data Mean = 122.3
Median = 77.5
Substantially different

154. Notions of Center E.g. British Suicides Data Mean = 122.3
Median = 77.5
Substantially different
But which is better?

155. Notions of Center E.g. British Suicides Data Mean = 122.3
Median = 77.5
Substantially different
But which is better?
Goal 1: ½ - ½ middle

156. Notions of Center E.g. British Suicides Data Mean = 122.3
Median = 77.5
Substantially different
But which is better?
Goal 1: ½ - ½ middle
Goal 2: long run average