1. Last Time Proportions Continuous Random Variables Probabilities
Mean and Standard Deviation
Male-Female Example
Continuous Random Variables
Uniform Distribution
Normal Distribution

2. Reading In Textbook Approximate Reading for Today’s Material:
Pages 58-64,
Approximate Reading for Next Class:
Pages 66-70, 61-62, 59-61,

3. Proportions Summary: Simple pattern
But be careful to keep these straight!
Counts, X
Proportions
Expected Value np p
Variance
np(1-p)
p(1-p)/n
Standard Deviation
(np(1-p))1/2
(p(1-p)/n)1/2

4. Big Picture Now exploit constant shape property of Binom’l
Centerpoint feels p Spread feels n
Big Question: What is common shape?

5. Normal Distribution
Continuous f(x),

6. Normal Distribution Continuous f(x),
that models common shape seen above

7. Normal Distribution Continuous f(x),
that models common shape seen above
Goal:
Shaped like a mound

8. Normal Distribution Continuous f(x),
that models common shape seen above
Goal:
Shaped like a mound
E.g. sand dumped from a truck

9. Normal Distribution Continuous f(x),
that models common shape seen above
Goal:
Shaped like a mound
E.g. sand dumped from a truck
Older (worse) description: bell shaped

10. Normal Distribution Continuous f(x),
that models common shape seen above
Like Binomial is indexed family of dist’ns

11. Normal Distribution Continuous f(x),
that models common shape seen above
Like Binomial is indexed family of dist’ns
Indexed by parameters

12. Normal Distribution Continuous f(x),
that models common shape seen above
Like Binomial is indexed family of dist’ns
Indexed by parameters
μ = mean (average, i.e. center)
Greek “mu”

13. Normal Distribution Continuous f(x),
that models common shape seen above
Like Binomial is indexed family of dist’ns
Indexed by parameters
μ = mean (average, i.e. center)
σ = standard deviation (spread)

14. Normal Distribution Nice insight into effect of μ and σ:
Applet by Webster West:

15. Normal Distribution Nice insight into effect of μ and σ:
Applet by Webster West:

16. Normal Distribution Nice insight into effect of μ and σ:
Applet by Webster West:

17. Normal Distribution
The “normal density curve” is:
usual “function” of

18. Normal Distribution The “normal density curve” is:
circle constant = 3.14…

19. Normal Distribution The “normal density curve” is:
natural number = 2.7…

20. Normal Curve Mathematics
Main Ideas:
Basic shape is:

21. Normal Curve Mathematics
Main Ideas:
Basic shape is:
“Shifted to mu”:

22. Normal Curve Mathematics
Main Ideas:
Basic shape is:
“Shifted to mu”:
“Scaled by sigma”:

23. Normal Curve Mathematics
Main Ideas:
Basic shape is:
“Shifted to mu”:
“Scaled by sigma”:
Make Total Area = 1: divide by

24. Normal Curve Mathematics
Main Ideas:
Basic shape is:
“Shifted to mu”:
“Scaled by sigma”:
Make Total Area = 1: divide by
as , but never

25. Normal Distribution
The “normal density curve” is:

26. Normal Distribution The “normal density curve” is:
Application: fit to data

27. Normal Distribution The “normal density curve” is:
Application: fit to data
i.e. Choose μ and σ to fit to histogram

28. Normal Density Fitting
Idea: Choose μ and σ to fit curve
to histogram of data,

29. Normal Density Fitting
Idea: Choose μ and σ to fit normal density
to histogram of data,

30. Normal Density Fitting
Idea: Choose μ and σ to fit normal density
to histogram of data,
Approach:
IF the distribution is “mound shaped”

31. Normal Density Fitting
Idea: Choose μ and σ to fit normal density
to histogram of data,
Approach:
IF the distribution is “mound shaped”
& outliers are negligible

32. Normal Density Fitting
Idea: Choose μ and σ to fit normal density
to histogram of data,
Approach:
IF the distribution is “mound shaped”
& outliers are negligible
THEN a “good” choice of normal model is:

33. Normal Density Fitting
Idea: Choose μ and σ to fit normal density
to histogram of data,
Example:
Melbourne Average Temperature Data

34. Normal Density Fitting
Idea: Choose μ and σ to fit normal density
to histogram of data,
Example:
Melbourne Average Temperature Data
Major City in Australia

35. Normal Density Fitting
Idea: Choose μ and σ to normal density
to histogram of data,
Example:
Melbourne Average Temperature Data
Major City in Australia

36. Normal Density Fitting
Idea: Choose μ and σ to normal density
to histogram of data,
Example:
Melbourne Average Temperature Data
Major City in Australia

37. Normal Density Fitting
Idea: Choose μ and σ to fit normal density
to histogram of data,
Example:
Melbourne Average Temperature Data
Major City in Australia, on Arctic Ocean

38. Normal Density Fitting
Idea: Choose μ and σ to fit normal density
to histogram of data,
Example:
Melbourne Average Temperature Data
Major City in Australia, on Arctic Ocean
Study winter temperatures

39. Normal Density Fitting
Idea: Choose μ and σ to fit normal density
to histogram of data,
Example:
Melbourne Average Temperature Data
Major City in Australia, on Arctic Ocean
Study winter temperatures
Averaged over

40. Normal Density Fitting
Idea: Choose μ and σ to fit normal density
to histogram of data,
Example:
Melbourne Average Temperature Data
Analyzed in:

41. Normal Density Fitting
Melbourne Average Temperature Data
Data in degrees
Centigrade
(world standard)

42. Normal Density Fitting
Melbourne Average Temperature Data
Data in degrees
Centigrade
(world standard)
Convert to
Fahrenheit
(interpretable
for us)

43. Normal Density Fitting
Melbourne Average Temperature Data
Data in degrees
Centigrade
(world standard)
Convert to
Fahrenheit
(interpretable
for us)

44. Normal Density Fitting
Melbourne Average Temperature Data
Restrict Attention to Winter Only

45. Normal Density Fitting
Melbourne Average Temperature Data
Restrict Attention to Winter Only
Since this looks normal
(recall mound shaped, no outliers)

46. Normal Density Fitting
Melbourne Average Temperature Data
Restrict Attention to Winter Only
Since this looks normal
(recall mound shaped, no outliers)
Will study summer later
(needs more powerful methods)

47. Normal Density Fitting
Melbourne Average Temperature Data
Histogram (for
winter only):

48. Normal Density Fitting
Melbourne Average Temperature Data
Histogram (for
winter only):
Mound shaped

49. Normal Density Fitting
Melbourne Average Temperature Data
Histogram (for
winter only):
Mound shaped
No outliers

50. Normal Density Fitting
Melbourne Average Temperature Data
Histogram (for
winter only):
Mound shaped
No outliers
So fit Normal
density

51. Normal Density Fitting
Melbourne Average Temperature Data
Fit Normal density

52. Normal Density Fitting
Melbourne Average Temperature Data
Fit Normal density

53. Normal Density Fitting
Melbourne Average Temperature Data
Fit Normal density
Use from data:
- Mean

54. Normal Density Fitting
Melbourne Average Temperature Data
Fit Normal density
Use from data:
- Mean
- S. D.

55. Normal Density Fitting
Melbourne Average Temperature Data
Fit Normal density
- Create Grid

56. Normal Density Fitting
Melbourne Average Temperature Data
Fit Normal density
- Create Grid
(usual type 1st two.
highlight & drag
corner)

57. Normal Density Fitting
Melbourne Average Temperature Data
Fit Normal density
- Create Grid
- Can compute directly
from density formula

58. Normal Density Fitting
Melbourne Average Temperature Data
Fit Normal density
- Create Grid
- Can compute directly
from density formula
Using x, μ, σ, etc.

59. Normal Density Fitting
Melbourne Average Temperature Data
Fit Normal density
- Create Grid
- Can compute directly
from density formula
- But faster to use
NORMDIST

60. Normal Density Fitting
Melbourne Average Temperature Data
Fit Normal density
- Create Grid
- Can compute directly
from density formula
- But faster to use
NORMDIST
(parallels BINOMDIST)

61. Normal Density Fitting
Melbourne Average Temperature Data
Use NORMDIST
Inputs:
- Xgrid

62. Normal Density Fitting
Melbourne Average Temperature Data
Use NORMDIST
Inputs:
- Xgrid
- Data mean

63. Normal Density Fitting
Melbourne Average Temperature Data
Use NORMDIST
Inputs:
- Xgrid
- Data mean
- Data S. D.

64. Normal Density Fitting
Melbourne Average Temperature Data
Use NORMDIST
Inputs:
- Xgrid
- Data mean
- Data S. D.
- Not Cumulative

65. Normal Density Fitting
Melbourne Average Temperature Data
Use NORMDIST
Use $ signs
for correct drag
& drop

66. Normal Density Fitting
Melbourne Average Temperature Data
Use NORMDIST
Use $ signs
for correct drag
& drop
- this changes

67. Normal Density Fitting
Melbourne Average Temperature Data
Use NORMDIST
Use $ signs
for correct drag
& drop
- this changes
- these are stable

68. Normal Density Fitting
Melbourne Average Temperature Data
Fit Normal density
Plot using:
- Insert

69. Normal Density Fitting
Melbourne Average Temperature Data
Fit Normal density
Plot using:
- Insert
- Line

70. Normal Density Fitting
Melbourne Average Temperature Data

71. Normal Density Fitting
Melbourne Average Temperature Data
Mounds shape is good visual fit to data

72. Normal Density Fitting
Melbourne Average Temperature Data
Mounds shape is good visual fit to data,
thus represents the population effectively

73. Normal Density Fitting
HW:
1.112

74. Melbourne Average Temperature Data
Research Corner
Melbourne Average Temperature Data
Mound shaped

75. Melbourne Average Temperature Data
Research Corner
Melbourne Average Temperature Data
Mound shaped
Or is it?

76. Melbourne Average Temperature Data
Research Corner
Melbourne Average Temperature Data
Mound shaped
Or is it?
Maybe 2 bumps?

77. Melbourne Average Temperature Data
Research Corner
Melbourne Average Temperature Data
Mound shaped
Or is it?
Maybe 2 bumps?
Or maybe 3?

78. Melbourne Average Temperature Data
Research Corner
Melbourne Average Temperature Data
Mound shaped
Or is it?
Maybe 2 bumps?
Or maybe 3?
Or explainable as
natural sampling
variation?

79. Melbourne Average Temperature Data
Research Corner
Melbourne Average Temperature Data
Explainable as
natural sampling
variation?

80. Melbourne Average Temperature Data
Research Corner
Melbourne Average Temperature Data
Explainable as
natural sampling
variation?
Useful tool:
Hypothesis Test

81. Melbourne Average Temperature Data
Research Corner
Melbourne Average Temperature Data
Explainable as
natural sampling
variation?
Useful tool:
Hypothesis Test
(will develop later,
need more background)

82. Normal Distribution
The “normal density curve” is:

83. Normal Distribution The “normal density curve” is:
Main Application: Probabilities
computed as areas under curve

84. (continuous probability density)
Normal Distribution
The “normal density curve” is:
Main Application: Probabilities
computed as areas under curve
(continuous probability density)

85. Computation of Normal Areas
E.g. for

86. Computation of Normal Areas
E.g. for
(center)

87. Computation of Normal Areas
E.g. for
(spread)

88. Computation of Normal Areas
E.g. for
(spread)
(# spaces: mean to inflection points)

89. Computation of Normal Areas
E.g. for
Area below 1.3
(point on # line)

90. Computation of Normal Areas
E.g. for
Area below 1.3 = ?

91. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?

92. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?
Calculus?

93. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?
Calculus?

94. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?
Calculus?

95. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?
Calculus?
Hurdle: no elementary anti-derivative

96. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?
Calculus?
Hurdle: no elementary anti-derivative
i.e. approaches of subst., parts, … all fail

97. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?

98. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?
Alternate Approach:
Numerical Methods

99. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?
Alternate Approach:
Numerical Methods
(e.g. Riemann summation)

100. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?
Alternate Approach:
Numerical Methods
Studied in course on Numerical Analysis

101. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?
Convenient Shortcuts

102. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?
Convenient Shortcuts (summarized answers):

103. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?
Convenient Shortcuts (summarized answers):
Historical: table (see Table A in text)

104. Computation of Normal Areas
E.g. for
Area below 1.3 = ?
How to compute?
Convenient Shortcuts (summarized answers):
Historical: table (see Table A in text)
Excel: function NORMDIST

105. Computation of Normal Areas
EXCEL function
NORMDIST:
works in terms of “lower areas”
E.g. for
Area ≤ 1.3

106. Computation of Normal Areas
EXCEL function
NORMDIST:
works in terms of “lower areas”
E.g. for
Area ≤ 1.3 Mean = 1

107. Computation of Normal Areas
EXCEL function
NORMDIST:
works in terms of “lower areas”
E.g. for
Area ≤ 1.3 Mean = 1
s.d. = 0.5

108. Computation of Normal Areas
EXCEL function
NORMDIST:
works in terms of “lower areas”
E.g. for
Area ≤ 1.3 Mean = 1
s.d. = 0.5

109. Computation of Normal Areas
EXCEL function
NORMDIST:
Many similarities to
BINOMDIST

110. Computation of Normal Areas
EXCEL function
NORMDIST:
works in terms of “lower areas”
E.g. for
Area ≤ 1.3 Mean = 1
s.d. = 0.5

111. Computation of Normal Areas
EXCEL function
NORMDIST:
works in terms of “lower areas”
E.g. for
Area ≤ 1.3 Mean = 1
s.d. = 0.5

112. Computation of Normal Areas
EXCEL function
NORMDIST:
works in terms of
“lower areas”
E.g. for
Area ≤ 1.3 Mean = 1
s.d. = 0.5

113. Computation of Normal Areas
EXCEL function
NORMDIST:
works in terms of
“lower areas”
E.g. for
Area ≤ 1.3
Mean = 1
s.d. = 0.5 Set “Cumulative” to true

114. Computation of Normal Areas
EXCEL function
NORMDIST:
works in terms of “lower areas”
E.g. for
Area ≤ 1.3 = = 0.726

115. Computation of Normal Probs
EXCEL Computation:
probs given by “lower areas”
E.g. for X ~ N(1,0.5)
P{X ≤ 1.3} = 0.726

116. Computation of Normal Probs
HW:
1.124
1.121a
Notes:
“Standard Normal” has mean 0, and s.d. 1
For “shade area”, use Excel to draw density, and then shade by hand

117. And Now for Something Completely Different
Fun Video
8 year old Skateboarding Twins:
Do they ever miss?
You can explore farther…
Thanks to previous student Devin Coley for the link

118. Computation of Normal Probs
EXCEL Computation:
probs given by “lower areas”
E.g. for X ~ N(1,0.5)
P{X ≤ 1.3} = 0.726

119. Computation of Normal Probs
EXCEL Computation:
probs given by “lower areas”
E.g. for X ~ N(1,0.5)
P{X ≤ 1.3} = 0.726
What about other probabilities?

120. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?

121. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
General Rules:
Express as “lower probs”

122. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
General Rules:
Express as “lower probs”
And use Big Rules of Probability
(Same spirit as BINOMDIST before)

123. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
Nice Visualization: StatsPortal
Resources
Statistical Applets
Normal Curve

124. Computation of Normal Probs
StatsPortal View
E.g. for
Area < 1.3
is 0.726

125. Computation of Normal Probs
Computation of upper areas:
(use “1 –”, i.e. “not” formula) =

126. Computation of Normal Probs
Computation of areas over intervals:
(use subtraction) =

127. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
Class Example 9:

128. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 1.3}
P{X ≥ 1.3}
P{-4 < X < 1.3}
P{X < 0.7 or X > 1.3}
P{|X – 1| > 0.3}
P{|X| > 0.8}

129. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 1.3}

130. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 1.3} = P{X ≤ 1.3}

131. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 1.3} = P{X ≤ 1.3}
(since P{X = 1.3} = 0)

132. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 1.3} = P{X ≤ 1.3}
(since P{X = 1.3} = 0)
(since X is a continuous random variable)

133. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 1.3} = P{X ≤ 1.3} = 0.726

134. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 1.3} = P{X ≤ 1.3} = 0.726

135. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 1.3}
P{X ≥ 1.3}
P{-4 < X < 1.3}
P{X < 0.7 or X > 1.3}
P{|X – 1| > 0.3}
P{|X| > 0.8}

136. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X ≥ 1.3}
P{-4 < X < 1.3}
P{X < 0.7 or X > 1.3}
P{|X – 1| > 0.3}
P{|X| > 0.8}

137. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X ≥ 1.3}

138. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X ≥ 1.3} = 1 – P{not(X ≥ 1.3)}

139. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X ≥ 1.3} = 1 – P{not(X ≥ 1.3)}
= 1 – P{X < 1.3}

140. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X ≥ 1.3} = 1 – P{not(X ≥ 1.3)}
= 1 – P{X < 1.3}
= 1 – = 0.274

141. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X ≥ 1.3} = 1 – P{not(X ≥ 1.3)}
= 1 – P{X < 1.3}
= 1 – = 0.274

142. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X ≥ 1.3}
P{-4 < X < 1.3}
P{X < 0.7 or X > 1.3}
P{|X – 1| > 0.3}
P{|X| > 0.8}

143. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{-4 < X < 1.3}
P{X < 0.7 or X > 1.3}
P{|X – 1| > 0.3}
P{|X| > 0.8}

144. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{-4 < X < 1.3}

145. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{-4 < X < 1.3} = P{X < 1.3} – P{X < -4}

146. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{-4 < X < 1.3} = P{X < 1.3} – P{X < -4} =

147. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{-4 < X < 1.3} = P{X < 1.3} – P{X < -4} =

148. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{-4 < X < 1.3} = P{X < 1.3} – P{X < -4} =
Note: same answer as above

149. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{-4 < X < 1.3} = P{X < 1.3} – P{X < -4} =
Note: same answer as above Why?

150. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{-4 < X < 1.3} = P{X < 1.3} – P{X < -4} =
Note: same answer as above Why?
P{X < -4} ≈ 0 (to 4 decimal places)

151. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{-4 < X < 1.3}
P{X < 0.7 or X > 1.3}
P{|X – 1| > 0.3}
P{|X| > 0.8}

152. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3}
P{|X – 1| > 0.3}
P{|X| > 0.8}

153. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3}

154. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} =
= P{X < 0.7} + P{X > 1.3}

155. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} =
= P{X < 0.7} + P{X > 1.3}
= P{X < 0.7} P{X > 1.3}

156. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} =
= P{X < 0.7} + P{X > 1.3}
= P{X < 0.7} P{X > 1.3} =

157. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} =
= P{X < 0.7} + P{X > 1.3}
= P{X < 0.7} P{X > 1.3} =

158. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
Alternate approach:
use symmetry

159. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
Alternate approach:
use symmetry

160. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
= 2 * P{X < 0.7}

161. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
= 2 * P{X < 0.7}
= 0.549

162. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
= 2 * P{X < 0.7}
= 0.549

163. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
P{|X – 1| > 0.3}
P{|X| > 0.8}

164. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
P{|X – 1| > 0.3}

165. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
P{|X – 1| > 0.3}
Visualize on number line:

166. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
P{|X – 1| > 0.3}
Visualize on number line:

167. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
P{|X – 1| > 0.3}
Interval centered at 1

168. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
P{|X – 1| > 0.3}
# (spaces from 1) = 0.3

169. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
P{|X – 1| > 0.3}
# (spaces from 1) = 0.3
Thus same answer as above

170. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{X < 0.7 or X > 1.3} = 0.549
P{|X – 1| > 0.3}
P{|X| > 0.8}

171. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{|X| > 0.8}

172. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{|X| > 0.8}
Use symmetry here?

173. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{|X| > 0.8}
Use symmetry here?
Careful, asymmetric normal dist

174. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{|X| > 0.8} = P{X < -0.8 or X > 0.8}

175. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{|X| > 0.8} = P{X < -0.8 or X > 0.8}
= P{X < -0.8} + 1 – P{X < 0.8}

176. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{|X| > 0.8} = P{X < -0.8 or X > 0.8}
= P{X < -0.8} + 1 – P{X < 0.8}
= 0.656

177. Computation of Normal Probs
EXCEL Computation:
E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726
What about other probabilities?
P{|X| > 0.8} = P{X < -0.8 or X > 0.8}
= P{X < -0.8} + 1 – P{X < 0.8}
= 0.656

178. Computation of Normal Probs
HW:
1.125,
1.121 b, c, d
1.136 (0.079, 0.271)
1.137