© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
Symmetric Relations and Geometric Characterizations about Standard Normal Distribution
by Circle and Square
1
Pearson’s finding probability point:
Its cumulative distribution probability:
Kelley’s formulation as 27 percent rule:
Standard Normal Distribution (Aspect Ratio: , )
Please remember the following values.
E va lu at io ns
Random Walking
Shingo NAKANISHI (Osaka Inst. of Tech., Japan) Masamitsu OHNISHI (Osaka Univ., Japan)
𝟎. 𝟔𝟏𝟐𝟎𝟎𝟑
−𝟎. 𝟔𝟏𝟐𝟎𝟎𝟑
Grouping such as the proposal by Cox
Good
Bad
Middle
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
1. Reasons why Pearson’s finding probability point, 0.612003, is important.
2. Symmetric and Geometric Proposals
of Two types of Differential Equations
between Standard Normal Distribution and Inverse Mills Ratio 3. These drawing methods with Circle and Square
between Winners, Losers, and their Banker .
Aims and Viewpoints about Our Research
Nakanishi’s Website:
http://www.oit.ac.jp/center/~nakanishi/english/
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
𝑋 𝛼 ,𝑡
3
Changing the fee under the condition
Number of trial: 𝑡
0
𝐸 (𝑌 𝛼 ,𝑡 )
The fee is equal to （ ） times of
based on Ref. TORSJ，55,1-26（2012）
Winners Maximal Profit are equilibrium to its Fee by a Banker at t=1 and s=1
Profit of a Winner :
𝑌 𝛼, 𝑡 = max(𝑋 𝛼, 𝑡 , 0)
Loss of a Loser :
𝑌 𝛼, 𝑡 = min(𝑋 𝛼, 𝑡 , 0)
Number of trial: 𝑡
Dashed curve is a winner’s expected profit which is greater than 0 : The maximal profit
for winners is a parabola based on the winners probability 0.2702678 constantly.
Equilibrium formulation :
If
and ,
Integral from 0 to ∞ of PDF
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
𝑈 𝑡 = 𝜆Φ −𝜆 𝑡
𝑈 𝑡 = 𝜆(1 + Φ −𝜆 ) 𝑡
𝑉 𝑡 = 𝜆 Φ −𝜆
Φ −𝜆 𝑡 = 𝜆 𝑡
𝑉 𝑡 = 𝜆 1 + Φ −𝜆
1 − Φ −𝜆 𝑡 𝛼𝑡
𝜆 𝑡 (Aspect Ratio:
𝟏. 𝟎)
Ref. TORSJ，55,1-26（2012）
Ref. ORSJ(@Kansai Univ.
in Sep., 2017)
Risk Aversion
Risk Preference
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
5
Relations between Inverse Mills Ratio, Conditional Expectation, and
1 2𝜋 2 2𝜋
u*=0.30263084
𝑔 𝑢 = 𝜙 𝑢 Φ 𝑢
𝑔 𝑢 = 𝜙 𝑢 Φ −𝑢
Bernoulli
differential equations
Ref. RIMS2078-10（@Kyoto Univ. in Nov., 2017）
Ref. ORSJ （ @Keio Univ. in Mar., 2016 ）
Kelley’s 27 percent rule, , shows
the conditional expected values:
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
Inverse Mills Ratio, Standard Normal Distribution, and Bernoulli Differential Equations
Ref. RIMS2078-10（Kyoto Univ. in Nov., 2017）
Kelley’s 27 percent rule,
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
− 1
𝜆 𝑢 + 1
𝜆Φ −𝜆 𝑣 = 1
− 1
𝜆 𝑢 + 1
𝜆 𝑣 = 1
− 1
𝜆 1 + Φ −𝜆 1 − Φ −𝜆
𝑢 + 1
𝜆 1 + Φ −𝜆 𝑣 = 1
: Entire Viewpoint （ Vertical axis ）
: Individual Viewpoint （ Horizontal axis ）
Intercept form of a linear equation for winners :
Ref. ORSJ（@Kansai Univ. in Sep., 2017）
Intercept form of a linear equation for losers :
Intercept form of a linear equation for their banker :
Standard Normal Distribution
by Circle and Square
for Winners, Losers,
and their Banker
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
Ref. RIMS Modified Version 2078-10
（See Research Gate 2019）
Ref. ORSJ Workshop(@National Graduate Institute for Policy Studies, in Nov., 2018.)
Variable coefficient type Second-Order Linear Differential Equations
about Integrals of CDF of
Standard Normal
Distribution
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
9
Integrals of CDF
Inverse Mills Ratio Negative
Inverse Mills Ratio Truncated
Normal Distribution
Modified Intercept forms
of linear equations
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
1. 𝑘 = ±0.0
2. 𝑘 = ±0.30263084 3. 𝑘 = ±0.506054
4. 𝑘 = ±0.612003(= 𝜆) 5. 𝑘 = ±0.67449
− 1 𝜙 𝑘 𝛷 −𝑘
𝑢 + 1
𝜙 𝑘 𝑣 = 1
− 1 𝜙 𝑘 𝛷 𝑘
𝑢 + 1
𝜙 𝑘 𝑣 = 1 Modified intercept forms
of linear equations according to 𝑘 𝑢
𝑣
0
𝑘 =Probability points Integrals of CDF
Inverse Mills Ratio Negative
Inverse Mills Ratio Truncated
Normal Distribution
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
11
1. 𝑘 = ±0.0
2. 𝑘 = ±0.30263084 3. 𝑘 = ±0.506054
4. 𝑘 = ±0.612003(= 𝜆) 5. 𝑘 = ±0.67449
− 1 𝜙 𝑘 𝛷 −𝑘
𝑢 + 1
𝜙 𝑘 𝑣 = 1
− 1 𝜙 𝑘 𝛷 𝑘
𝑢 + 1
𝜙 𝑘 𝑣 = 1 𝑢
𝑣
0
Modified intercept forms
of linear equations according to 𝑘
𝑘 =Probability points Integrals of CDF
Inverse Mills Ratio Negative
Inverse Mills Ratio Truncated
Normal Distribution
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
1. 𝑘 = ±0.0
2. 𝑘 = ±0.30263084 3. 𝑘 = ±0.506054
4. 𝑘 = ±0.612003(= 𝜆) 5. 𝑘 = ±0.67449
− 1 𝜙 𝑘 𝛷 −𝑘
𝑢 + 1
𝜙 𝑘 𝑣 = 1
− 1 𝜙 𝑘 𝛷 𝑘
𝑢 + 1
𝜙 𝑘 𝑣 = 1 𝑢
𝑣
0
Modified intercept forms
of linear equations according to 𝑘
𝑘 =Probability points Integrals of CDF
Inverse Mills Ratio Negative
Inverse Mills Ratio Truncated
Normal Distribution
The proportion :
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
13
1. 𝑘 = ±0.0
2. 𝑘 = ±0.30263084 3. 𝑘 = ±0.506054
4. 𝑘 = ±0.612003(= 𝜆) 5. 𝑘 = ±0.67449
− 1 𝜙 𝑘 𝛷 −𝑘
𝑢 + 1
𝜙 𝑘 𝑣 = 1
− 1 𝜙 𝑘 𝛷 𝑘
𝑢 + 1
𝜙 𝑘 𝑣 = 1 𝑢
𝑣
0
Modified intercept forms
of linear equations according to 𝑘
𝑘 =Probability points Integrals of CDF
Inverse Mills Ratio Negative
Inverse Mills Ratio Truncated
Normal Distribution
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
1. 𝑘 = ±0.0 ∵ Φ −𝑘 = Φ 𝑘 = 1/2 2. 𝑘 = ±0.30263084 ∵ 𝜙 𝑘 /Φ −𝑘 = 1 3. 𝑘 = ±0.506054 ∵ 𝑘 = 𝜙(𝑘)/Φ(𝑘) 4. 𝑘 = ±0.612003 ∵ 𝜙 𝑘 = 2𝑘Φ(−𝑘)
5. 𝑘 = ±𝟎. 𝟔𝟕𝟒𝟒𝟗 ∵ Φ −𝑘 = 1/4, Φ 𝑘 = 3/4
3 3
1 1 3
4 5
25%
25% : 75%=1/4 : 3/4
− 1 𝜙 𝑘 𝛷 −𝑘
𝑢 + 1
𝜙 𝑘 𝑣 = 1
− 1 𝜙 𝑘 𝛷 𝑘
𝑢 + 1
𝜙 𝑘 𝑣 = 1
Pythagorean Triangles show the probability as CDF.
ℎ 𝑢 = 𝜙 𝑢 + 𝑢𝛷 𝑢 ,
𝑔 𝑢 = 𝜙 𝑢 𝛷 𝑢
𝑑𝑔 𝑢
𝑑𝑢 + 𝑢𝑔 𝑢 + 𝑔 𝑢 = 0
𝑑 ℎ 𝑢
𝑑𝑢 + 𝑢 𝑑ℎ 𝑢
𝑑𝑢 −ℎ 𝑢 = 0
Modified intercept forms
of linear equations according to 𝑘
𝑘 =Probability points Integrals of CDF
Inverse Mills Ratio Negative
Inverse Mills Ratio Truncated
Normal Distribution
© Shingo Nakanishi (Osaka Institute of Technology, JAPAN)
15
Concluding Remarks
1. Importance of
Pearson’s finding probability point, 0.612003.
2. Symmetric Relations and Geometric Characterizations about Two types of Differential Equations
between Standard Normal Distribution and Inverse Mills Ratio.
3. Greek Pythagorean Theorem about CDF
and Ancient Egyptian Drawing Styles with Circle and Square.
4. Proposals of Modified Intercept Forms
for Winners, Losers, and Their Banker .
Acknowledgments
We would like to express our sincerely gratitude to Prof. Kosuke OYA and Prof. Hisashi TANIZAKI belonging to the Graduate School of Economics at Osaka University. The first author, Shingo NAKANISHI, would like to show my grateful to Prof. Hidemasa YOSHIMURA, Associate Prof. Manami SATO,
Prof. Tsuneo ISHIKAWA and Prof. Yukimasa MIYAGISHI belonging to Osaka Institute of Technology. And the first author would particularly like to thank Prof. Takeshi KOIDE at Konan University, Associate Prof. Hitoshi HOHJO at Osaka Prefecture University, Prof. Shoji KASAHARA at Nara Institute of Science and Technology, Prof. Jun KINIWA at University of Hyogo. Especially, we would like to show full of our appreciations to Prof. Tetsuya TAKINE belonging to the Graduate school of Engineering at Osaka University, Prof. Hiroaki SANDOH at Kwansei Gakuin University and many other members at Operations Research Society of Japan (ORSJ) and Kansai-tiku Koryukai at the Securities Analysts Association of Japan (SAAJ).