5-16-1-1 四角形や三角形の面積 1-①5-16-1-2 四角形や三角形の面積 1-② ₁ 次のような平行四辺形の面積を求めましょう。 （式10 点，答10
リスクとリターンが語る円・正方形・直角三角形と...
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Shingo NAKANISHI at Osaka Institute of Technology, Japan, Masamitsu OHNISHI at Osaka University, Japan. © 2019, 2-5, September, Belfast, UK, RSS (Royal Statistical Society) Annual Conference 2019 © 2019, Poster Presentation and Design: Shingo NAKANISHI, Osaka Institute of Technology, Japan Acknowledgements: The first author, Shingo NAKANISHI, would like to show my grateful to many research members at Operations Research Society of Japan, Kansai-tiku Koryukai of The Securities Analysts Association of Japan, Osaka Institute of Technology in Japan, and Osaka University in Japan. Original References: Operations Research Society of Japan (ORSJ), Research Institute for Mathematical Sciences, Kyoto University (RIMS Kôkyûroku, 2078-10), The 15th International Symposium on Econometric Theory and Applications (SETA2019). URL: http://www.oit.ac.jp/center/~nakanishi/english/ Geometric Characterizations and Symmetric Relations between Standard Normal Distribution and Inverse Mills Ratio based on Pythagorean Theorem 2 1 1 1 1 2 1 1 1 2 4 3 3 2 2 4 4 4 1 1 1 1 1 Slopes as Probabilities of standard normal distribution are connected to the right triangle by Pythagorean theorem geometrically. 1.0 x − x Probability point: 0 x = Probability point: 0.30263084( ) (() ( )) ( )/ ( ) 1 x x x x x η φ Φ φ Φ = = = − − = α α α β β Probability point: 0.506054 ( ( )/ ( )) () 0.693591 ( ) 0.306409 ( ) () () ( )/ () x x x x x x x x x x x x x x φ Φ Φ Φ α Φ β Φ φ φ Φ α β = = = − = = − = = = = = + 1 2 1 Probability point : 0 () 1 ( ) 0.5 ( ) 0.5 Proportion of right triangle=1:2: 5 x x x x Φ Φ Φ = = − − = − = ( ): () 0.5 : 0.5 1:1 x x Φ Φ − = = Probability point : 0.67449 () 1 ( ) 0.75 ( ) 0.25 Proportion of right triangle=3:4:5 x x x x Φ Φ Φ = = − − = − = ( ): () 0.25 : 0.75 1:3 x x Φ Φ − = = All points whithin the circle and square are converging at one point practically because their pobability points are 0. 1 t = 0 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 Green dashed lines are probability density functions (PDF): f(x), Red dashed lines are cumulative distribution functions (CDF): F(x), Purple 1-dot chain lines are integrals of cumulative distribution functions, Red dotted lines are 1.0, Purple dotted lines are asymptotic lines whose slopes are 1.0. (a) Type of a triangular distribution (b) Type of a uniform distribution (c) Type of an exponential distribution (d) Type of a normal distribution First concepts are shown as integral forms of cumulative distribution functions. 0, ( 1) 1, (1 0) () 1, (0 1) 0, ( 1) x x x fx x x x ≤− + −< ≤ = − + < ≤ > 0, ( 0) () 1, (0 1) 0, ( 1) x fx x x < = ≤ ≤ > 0, ( 0) () exp( ), ( 0) x fx x x < = − ≥ Aspect ratio 1.0 = F(x) F(x) F(x) F(x) - 40 - 20 0 20 40 -4 -3 -2 -1 0 1 t 1 2 1 2 1 , 1 , 0.5, 1.0 ( 0.10, 0.12, 0.14,0.16, ,0.38, 0.40) c c p p v α α α = − =− − = = = = 0.10 α = 0.12 α = 0.14 α = - 40 - 20 0 20 40 -6 -4 -2 0 2 4 t 1 2 1 2 1 , 1 , 0.5, 1.0 ( 0.10, 0.12, 0.14,0.16, ,0.38, 0.40) c c p p v α α α = − =− − = = = = 0.10 α = 0.12 α = 0.14 α = If we think of the repetition game of coin tossing with a constant fee, we show you two parabolic curves of the performances both winners and losers. We find that the maximal profit of winners is equal to its fee based on 27 percent probability. 0 () yu u λ λ λ − λ − 2 λ Interval of solid green line 0.777 ≈ Regression line: () yu u λ = Two dimensional standard normal distribution with correlation 2 1 1 () exp , ( ) 2 2 fx x x π = − −∞ < <∞ () (() ( )) ( ) Winners risk aversions W U t t t φλ λ λ λ λ = − Φ− = Φ− () (1 ( )) L U t t λ λ =− +Φ − * () (1 ( )) ( ) Losers risk preferences L U t t λ λ =− +Φ − − ( ) ( ) its fee ( ) ( ) W t V t t t λ λ λ α λ Φ− = = = = Φ− (1 ( )) () (1 ( )) (1 ( )) (1 ( )) L t V t t λ λ λ λ λ λ +Φ − =− −Φ − +Φ − =− −Φ − * (1 ( )) () ( ) (1 ( )) L V t t λ λ λ +Φ − =− − −Φ − ( ( , , )) W EY t ασ ( ( , , )) L EY t ασ ( , ,) W Z t ασ ( , ,) L Z t ασ 0 1 t = 0 1 t Positive expected profits 0 () Equilibrium point of inverse Mills ratio : 2 ( ) u φ λ λ = Φ− 0.612003 λ = Maximal loss out of ( ) is equal to its profit : λ λ Φ− Pr( () 0) ( ()| () 0) ( ) Xt EXt Xt λ < < Φ− 2 (1) ~ ( ,1 ) U X N λ λ + = Relations both squares and normal distribution based on 2λ t λ () () () u u u ψ φ Φ Investor’s profits and losses : () Xt Profits ( () 0) Xt > Losses ( () 0) Xt < Losses ( () 0) Xt < Profits ( () 0) Xt > Investor’s profits and losses : () Xt 0 Expectation of the truncated normal distribution is 2λ − 2λ λ λ λ PDF of the truncated normal distribution: () , ( ) ( ) () 0, ( ) u u u u φ λ λ ψ λ ≤− Φ− = >− λ 2 Case: 1, (1) ~ ( ,1 ) t X N λ = + Case: 2, (2) ~ ( 2 ,2) t X N λ = + Case: 3, (3) ~ ( 3 ,3) t X N λ = + Case: 4, (4) ~ ( 4 ,4) t X N λ = + 1 t Negative expected profits () () ( ) u u u ψ φ Φ− 0 Relations both squares and normal distribution based on 2λ Expectation of the truncated normal distribution is 2λ 2λ λ t λ − λ λ λ PDF of the truncated normal distribution: () , ( ) ( ) () 0, ( ) u u u u φ λ λ ψ λ ≥− Φ− = <− Maximal Profit out of ( ) is equal to its charge : λ λ Φ− 2 ()~ ( ,1 ) U Xt N λ λ − = − 2 Case: 1, (1) ~ ( ,1 ) t X N λ = − Case: 2, (2) ~ ( 2 ,2) t X N λ = − Case: 3, (3) ~ ( 3 ,3) t X N λ = − Case: 4, (4) ~ ( 4 ,4) t X N λ = − Pr( () 0) ( ()| () 0) ( ) Xt EXt Xt λ > > Φ− 0 0 0 If we consider 1 and 1, we can estimate the equilibrium point as . The integral results are as follows. () ( ( )) () () ( ( )) () ( ) ( ()~ ( , )), ( ) ( ( )) () ( t xtfxt dx t xtfxt dx t Xt N tt fxt dx t f σ µ λ λ λ λ λ λ ∞ −∞ −∞ = = =± −Φ− = =− + Φ− ∫ ∫ ∫ 0 ( ) ( ()~ ( , )). ( ) ( )) () Xt N tt xt dx t λ λ λ λ λ ∞ Φ− = = − Φ− ∫ If we think of a profit as when 1, we can use ()~ ( , ). Or if we think of a loss as when 1, we can use ()~ ( , ). Or if we think of them as 0 when 1, we can use ()~ (0, ). t t Xt N tt t t Xt N tt t Xt N t λ λ λ λ + = + − = − = 2 2 2 2 the interval of If we consider the solid green line i correlation coefficient as Karl Pear s nearly equal to 0 son's another finding value 0.612003, under the condition: () 2 () (1 .777 ). u yu uy u ρ λ λ λ + − = − Radius of the circle: ( 0.612003), Correlation coefficient: . λ ρ λ = = Relations between regression analysis, principal component analysis, and 0.612003. λ = ( ), ( ), ( ), , () P h u u u v u φ ψ Φ 0.612003 λ = 2λ λ 2λ − λ − λ − () u Φ () u φ u '' ' ' '' () () () 0 () () () () () () () P P P P P P h u uh u h u h u u u u h u u h u u φ φ + − = = + Φ =Φ = Self-adjoint second order differential equation of a standard normal distribution: ( ) λ λ Φ− (1 ( )) λ λ +Φ − 2 ( ) λ λ Φ− Tangent line of () ( ) () ( ) 2 () ( ) () () () 2 () ( ) P h u v u u u v u u u λ φλ λ λ λ λ λ φλ λ λ λ λ =Φ− + =Φ− + Φ =− =Φ + =Φ + Φ = () u Φ () u φ ( ), ( ), ( ), , () P g v v v u v φ ψ Φ 0. Background We present the geometric characterizations and symmetric relations between standard normal distribution and inverse Mills ratio by circles and squares from the viewpoint with considering the height of densities such as ancient Egyptian drawing styles and using the Greek Pythagorean Theorem. First, we can clarify the integral forms of various cumulative distribution functions including standard normal distribution based on the aspect ratio (=1.0, see 1). Second, we reconsider what the several times of standard deviations multiplied by the square root of the time mean with their positive and negative expectations multiplied by the time under the condition the aspect ratio=1.0 (see 2). At this time, we can understand the following things. (1) We can find the equilibrium formulation at any real numbers of the time t (see 2). (2) Its constant number, 0.612003, was found by Karl Pearson about 100 years ago. (3) Sir David Roxbee Cox reconfirmed the value to cluster the normal distribution. (4) Truman Lee Kelley also proposed the 27 percent rule formulation. Third, we can show the parabola for maximal profits including fees. And that is equal to its fee based on the 27 percent probability. In addition to these tendencies, we can clarify the relations between winners, losers, and their banker (see 3). Fourth, we can understand two types of ordinary differential equations to explain that with circles and squares (see 4). One is the Bernoulli differential equation of inverse Mills ratio. The other is the second order linear differential equation of the integral of cumulative normal distribution function (see 1). From these tendencies, we can also get the modified intercept forms geometrically and symmetrically for maximal profits of winners, these losses of losers, and their banker’s fee. We can understand that these equations should be changing the probability points and these probabilities based on Pythagorean Theorem correctly. If the probability point is that found by Karl Pearson, we can show you that it is the special point of standard normal distribution such as Sir Cox’s proposal. Other point on the right triangle with 3:4:5 is also the third quantile of standard normal distribution. Finally, we can also realize there are many similar tendencies close to the relations between circles and squares such as Vitruvian man by Da Vinci and various Mandalas although there might not be related to normal distribution directly and historically. The ancient Egyptian drawing styles enable us to illustrate the geometric characterizations and symmetric relations between standard normal distribution and inverse Mills ratio with circle and square based on the Pythagorean theorem in the ancient Greece. We think that our ideas shall be contributed in the statistical modelling and these evaluation fields since our suggested figures should be much more easily understood and powerful than we thought. ' 2 1 2 Bernouilli differential equation of an inverse Mills ratio : () () () 0 () () () () ( ) , ( 0, inverse Mills ratio) () () () , ( ( )) ( ) () P P P P P P g v ug v g v v g v v C v g v C v v g v C v φ φ φ λ λ + + = = Φ + = = Φ = =Φ− Φ− +Φ Equiliburium point between a banker, winners and losers : 0.612003 λ = () 2 ( ) φλ λΦ λ = − Modified intercept form of linear equation for winners : 1 1 1 () () ( ) Modified intercept form of linear equation for losers : 1 1 1 () () () Modified intercept form of linear equation for t u v u v φλ φλ λ φλ φλ λ − + = Φ− − + = Φ heir banker : 1 1 1 () () ( ) () u v φλ φλ λ λ − + = Φ− +Φ () v Φ 2 () v λΦ v () () () P h u u u u φ = + Φ () () () P h u u u u φ = + Φ λ λ − 2 () () ( ) () P v g v v φ λ = Φ− +Φ 1 () () () P v g v v φ = Φ () ( ) v λ Φ − Φ− Negative probability density function of truncated normal distribution at 0.612003 : () () ( ) v v λ φ ψ λ = − =− Φ− 1 () () () P v g v v φ = Φ 2 () () ( ) () P v g v v φ λ = Φ− +Φ () () ( ) v v φ ψ λ = Φ− Probability density function of truncated normal distribution at 0.612003 : () () ( ) u u λ φ ψ λ = = Φ− 2 () u λΦ 0 2λ − 1.0 − λ − λ λ λ − 0 v u = v u = v u = v u =− v u =− v u =− Pythagorean Triangles show the probabilities as CDFs. Inverse Mills Ratios and PDFs are discribed as Modified Intercept Forms. 2 2 2 2 2 0 2 2 2 2 ( ( , ,) ) 1 1 1 1(( , ,) ) ( , ,) exp ( , ,) ( , ,) exp ( , 2 2 ( ( , , )) 2 2 ( , ,) Pr( ( , ,) 0) ( ( , ,) ) 1 1 exp ( , ,) 2 2 W W W W W W W y t t x t t y t dy t x t dx t t EY t t t Z t X t y t t dy t t t ασ α ασ α ασ ασ ασ α σ σ ασ πσ πσ ασ ασ ασ α ασ σ πσ ∞ ∞ −∞ ∞ −∞ + + ⋅ − ⋅ − = = = > + − ∫ ∫ ∫ 2 2 0 2 2 2 2 2 2 2 ,) 1 1(( , ,) ) exp ( , ,) 2 2 ( ( , ,) ) 1 1 ( , ,) exp ( , ,) 2 ( ( , , )) 2 ( , ,) Pr( ( , ,) 0) ( ( , ,) ) 1 1 exp ( , ,) 2 2 L L L L L L L t x t t dx t t t y t t y t dy t x t EY t t Z t X t y t t dy t t t σ ασ α ασ σ πσ ασ α ασ ασ σ ασ πσ ασ ασ ασ α ασ σ πσ ∞ ∞ −∞ ∞ −∞ + − + ⋅ − = = = < + − ∫ ∫ ∫ 2 0 2 2 2 0 2 2 * 0 * * 1 1(( , ,) ) ( , ,) exp ( , ,) 2 2 1 1(( , ,) ) exp ( , ,) 2 2 ( , , ) max ( ( , , )) ( , , ) ( ( , , )) ( , ,) max[ ( , , ),0] ( , ,) min[ W W t L L W L x t t t dx t t t x t t dx t t t U t EY t U t EY t Y t X t Y t τ ασ α ασ ασ σ πσ ασ α ασ σ πσ ασ ασ ασ ασ ασ ασ ασ −∞ −∞ << + ⋅ − + − = = = = ∫ ∫ ( , , ),0] X t ασ Concluding Remarks We can understand that the slopes should be the probabilites of standard normal distribution based on the probability points. And above numbers are also the special probability points. Intercept form of linear equation for winners : 1 1 1 ( ) Intercept form of linear equation for losers : 1 1 1 (1 ( )) 1 ( ) 1 ( ) Intercept form of linear equation for their banker : u v u v λ λ λ λ λ λ λ λ − + = Φ− − + = +Φ − +Φ − −Φ − − 1 1 1 u v λ λ + = () (() ( )) N h u u u u φ − =− − Φ− () ( ) 0 ( ) () 0 () () ( ) ( ) P N P N P N P N h u h u h u h u h u h u u h u h u u − − = − − = − = − − − =− 1 1 ( () ( )) 2 2 P N h u h u u − = λ λ − 0 () u φ 0 2λ − λ − λ () (() ( )) N h u u u u φ − =− − Φ− 1 () () () P v g v v φ − =− Φ Negative probability density function of truncated normal distribution at 0.612003 : () () ( ) v v λ φ ψ λ = − =− Φ− 0.612003 λ = λ − () u Φ () () () P h u u u u φ = + Φ 2 () () ( ) () P v g v v φ λ = Φ− +Φ v u = v u =− ( ), ( ), ( ), , () P g v v v u v φ ψ Φ () v Φ 1 () () () P v g v v φ = Φ ( ), ( ), ( ), , () P h u u u v u φ ψ Φ ( ) λ λ Φ− 2 ( ) λ λ Φ− () v u λ =Φ ( ) v u λ =Φ− ( ) v u λ = −Φ − v u λ = + ( ) 2 ( ) v u λ λ λ =Φ− + Φ− () 2 ( ) v u λ λ λ =Φ − Φ− ( ) 2 ( ) v u λ λ λ =Φ− − Φ− 1 () () () ( ) () P N vu h u h u λ λ =Φ +Φ − 2 () ( ( ) () () ( )) P N v u h u h u λ λ − =−Φ− +Φ 1 2 1 2 () () () () () () (1 2 ( ))( () ( )) 2 ( ) P N P N vu v u h u h u vu v u h u h u u u λ λ + = + − = − Φ− − = − Φ− 1 2 If , then () () 2 ( ) u v v λ λ λ λ λ λ = − = − Φ− 1 () () ( ) N v g v v φ = Φ− 2 () () ( ) ( ) N v g v v φ λ = Φ− +Φ − () 2 ( ) φλ λ λ = Φ− () 2 ( ) φλ λ λ = Φ− () 2 ( ) φλ λ λ = Φ− 2 (0) φ − 4 (0) φ − λ λ 2 φλ λ λ ( )= Φ(− ) λ λ Φ(− ) λ λ (1 ) λ λ + Φ(− ) (1 ) λ λ − Φ(− ) 1.0 2λ 2 CDF of (0,1 ) : () N u Φ 2 PDF of (0,1 ) : () N u φ 0.0 PDF of truncated normal distribution: () , ( ) ( ) () 0, ( ) u u u u φ λ λ ψ λ ≤− Φ− = >− () () () u u u ψ φ Φ 1 ( ) 1 ( ) λ λ λ +Φ− −Φ− λ λ Φ(− ) λ λ (1 − Φ(− )) 1 3 =0.612004619 1 2 3 λ + ≅ + Our concept of the integration of standard normal disribution with a square and a circle is similar to “Mandala” and “Vitruvian Man” which is one of da Vinci’s works coincidentally. The original reference websites are https://en.wikipedia.org/wiki/Mandala, https://ja.wikipedia.org/wiki/ 曼荼羅, https://en.wikipedia.org/wiki/Vitruvian_Man (Access date: July, 3rd, 2017). Moreover, we are able to refer the idea and figure of Prof. Ida’s website which is discribed that the ratio should be less than the inverse number of the golden ratio. Reference: Ida, T., “Vitruvian Man by Leonardo da Vinci and the Golden Ratio” , http://www.crl.nitech.ac.jp/~ida/education/VitruvianMan/index.html (Access date: July, 3rd, 2017). We estimate the practical approximated value such as “Squaring the circle” which is simmilar to a problem proposed by ancient geometers. https://en.wikipedia.org/wiki/Squaring_the_circle (Access date: November 18th, 2017) u 1. First concept as integrals of various cumulative distributions 2. Second concept as distances of probability points based on the time t. There should be maximal losses against the positive returns or maximal profits against the negative returns by the 27 percent probabilities. 3. Relations between winners, losers, and their banker by the equilibrium formulation and 4. Main concept as the relations between Pythagorean theorem, differential equations, circles, and squares on standard normal distribution ( ) 0.2702678 λ Φ− = 0.612003 t α λ σ = = 5. Special case as the geometric characterizations and rotationally symmetric relations between winners, losers, and their banker based on the condition: and MP4 Visual Animation PowerPoint Visual Animation ( ) 0.2702678. λ Φ− = 0.612003 λ = That is, the fee is equal to . α λ 3:4:5 著者・作成者： 中西 真悟 （大阪工業大学情報センター・准教授） 共同研究者 ： 大西 匡光 （大阪大学大学院経済学研究科 & 数理・データ科学教育研究センター・教授） 謝 辞： ご助言，ご支援，ご協力を賜った多くの日本オペレーションズ・リサーチ学会員の皆様， 日本証券アナリスト協会関西地区交流会の皆様，大阪大学と大阪工業大学の教職員の皆様に厚く御礼を述べます． また，若い頃の恩師 中易秀敏 先生，栗山仙之助 先生と，リスクと確率論について手解きしてくれた 若い頃の上司 亀島鉱二 先生，志垣一郎 先生に感謝いたします． 掲示図の出典： 日本オペレーションズ・リサーチ学会論文と研究発表会資料 (ORSJ)， 京都大学数理解析研究所講究録 (RIMS Kôkyûroku, 2078-10) と修正版 , 第 15 回計量経済学の理論と応用に関する国際シンポジウムへの投稿論文とプレゼンテーション (SETA2019)， 2019 年王立統計学会年次大会ポスター・プレゼンテーション (RSS2019). リスクとリターンが語る円・正方形・直角三角形と 標準正規分布の対称的な幾何学的特性 URL: http://www.oit.ac.jp/center/~nakanishi/ 2019 年 9 月 13 日に 大阪工業大学梅田キャンパスで開催の イノベーションデイズにおいて 展示した作品をベースに作成 Version 2 注 1： 左側の白枠中は，大阪工業大学イノベーションデイズでの掲載作品です．一部は OR 学会 2019 年秋季研究発表会や 2019 年王立統計学会国際会議でも発表しています． 右側は 2019 年の王立統計学会主催の国際会議ポスタープレゼンテーションで発表した掲載作品です．円や曲線を含むポスターのデザインは大阪の夜空に打ち上がる「花火の色彩と背景」を表現しています． 注 2： ポスターの背景のデザインの色合いは本学のスクールカラー「紺青，大阪工大ブルー」，コミュニケーションマークのカラー「シアン」， 淀川や大阪湾も含めて世界へ発信できる「水の都 大阪」をイメージして作成しています． © 中西真悟（大阪工業大学），大西匡光（大阪大学），2019 勝者，敗者，胴元の均衡点: 0.612003 λ = () 2 ( ) φλ λΦ λ = − ( ), ( ), ( ), , () P h u u u v u φ ψ Φ v u = v u =− From 大阪工業大学大宮キャンパス 南東側：生駒山 北西側：六甲山 1 1 2 1 1 1 1 2 1 1 1 2 4 3 3 2 2 4 4 4 1 1 1 Probabilities of standard normal distribution are connected to the right triangle by Pythagorean theorm geometrically. 1 Probability point : 0 () 1 ( ) 0.5 ( ) 0.5 Proportion of right triangle=1:2: 5 x x x x Φ Φ Φ = = − − = − = ( ): () 0.5 : 0.5 1:1 x x Φ Φ − = = Probability point : 0.67449 () 1 ( ) 0.75 ( ) 0.25 Proportion of right triangle=3:4:5 x x x x Φ Φ Φ = = − − = − = ( ): () 0.25 : 0.75 1:3 x x Φ Φ − = = All points whithin the circle and square are converging at one point practically because their pobability points are 0. 勝者，敗者，胴元の均衡点: 0.612003 λ = () 2 ( ) φλ λΦ λ = − 標準正規分布に二階線形微分方程式と ベルヌーイ型微分方程式を用いた． 古代エジプトの絵画図の技法のように 奥行きを取り除いて回転させて重ね合 わせてみると，ピタゴラスの定理とパ ラメトリック方程式による円，正方形， 直角三角形の美しい幾何学模様が描ける． 緑色は 標準正規分布 による二階線 形微分方程式 に関連する曲線 赤色は 標準正規分布 （逆ミルズ比）に よるベルヌーイ 型微分方程式 に関連する曲 線 水色は 標準正規分布による 二階線形微分方程式を 用いたパラメトリック 方程式に関連する曲線 1 1 1 () () ( ) 1 1 1 () () () 1 1 1 () () ( ) () k k k k k k k k k k φ φ φ φ φ φ − + = Φ− − + = Φ − + = Φ− +Φ 勝者の切片系の方程式： 敗者の切片系の方程式： 胴元の切片系の方程式： 大阪の街から見える生駒山と六甲山の山の高さは どことなく縦横比が同じときの標準正規分布の高さに似ている． 同時に，歴史を通じた文化を感じながら曼荼羅のような幾何学模様を描ける． λ λ λ λ (0) φ ( ) λ λ Φ− 2 ( ) λ λ Φ− ( ) λ λ Φ− λ 1.0 2λ () u φ 0.0 1.0 () ( ) 2 ( ) fu u λ λ λ = −Φ − + Φ− () (1 ( )) 2 ( ) fu u λ λ λ =− −Φ− + Φ− 2 () () ( ) h u u u u φ = − Φ− () () ( ) u u φ ψ λ = Φ− u Tangential equation : 2 () ( ) dh u u du = −Φ − Probability density function of a trancated normal distribution : Probability density function of a standard normal distribution : Cumulative distribution function of a standard normal distribution : () u φ ( ) u Φ− φλ λ λ ( ) = 2 Φ(− ) Equiliburium point between a banker, winners and losers : 0.612003 λ = Intercept form of a linear equation for winners : Intercept form of a linear equation for losers : Intercept form of a linear equation for a banker : 2 2 2 1 1 1 1 1 1 (1 ) 1 1 1 1 1 u u u φ λ λ λ φ λ λ λ λ λ φ λ λ − + = Φ(− ) − + = + Φ(− ) + Φ(− ) − Φ(− ) − + = 2 2 () ( () ) () u fu h u u φ φ ψ 2 2 2 () () dh u u du φ = () 2 ( ) φλ λ λ = Φ− Equiliburium point of an inverse Mills ratio : Utility function for winners : () (() ( )) ( ) W U t t t φλ λ λ λ λ = − Φ− = Φ− ( ) ( ) 1 '' ' 2 2 2 ' 1 ' 1 2 2 Self-adjoint differential equation: () ( () () ( )) 0 () () () () 0 u h u uh u h u u h u u h u φ φ φ − − − + − = − = Variable coefficient second order linear homogeneous differential equation for a standard normal distribution : 2 2 2 2 2 2 2 () () () 0, 1 (0) ( (0)), 2 (0) 1 ( (0)) 2 dh u dh u u h u du du h dh du φ π + − = = = =− = −Φ (1 ( )) 2 () (2 0) () ( ) 2 () (0 2) u u fu u u λ λ λ λ λ λ λ λ − −Φ − + Φ − ≤ ≤ = −Φ − + Φ ≤ ≤ 2 u λΦ( ) u Φ( ) 2 2 Cumulative distribution f Probabilistic density funct unction of a standard norma ion of a standard normal distribution: 1 1 () exp 2 l distribution: 1 1 () exp 2 2 1 or ( ) e 2 2 u u u u du u u φ π π π −∞ Φ = − Φ − = = − ∫ 2 1 xp 2 u u du − −∞ − ∫ 2 2π 1 2π u φ λ () Φ(− ) λ − λ 0 λ λ Φ(− ) 2 λ λ φλ Φ(− )= ( ) u 2λ λ 1.0 u Φ( ) u λ 2 Φ( ) ( ) u u φ λ () Φ(− )+Φ − () u u φ λ () Φ(− )+Φ () u u φ () Φ ( ) u u φ () Φ− u λ 2 Φ(− ) u Φ(− ) max ( ) ( ) 2 ( ) max ( ) () 2 ( ) u u u u φ φλ λ λ λ φ φλ λ λ λ () ( ) = = Φ− +Φ− Φ− () ( ) = = Φ− +Φ Φ− If then C λ = Φ(− ) λ λ φλ λ λ = 0.612003 Φ(− ) = 0.2702678 ( ) = 2 Φ(− ) 0.30263084 0.3810856 η η φη η = Φ(− ) = ( ) = Φ(− ) Inverse Mills ratio known as hazard function: Survival ratio of a standard normal distribution: () fu u = () fu u =− Probabilistic density function of a trancated normal distribution: ( ,1.0) η 2 Bernoulli differential equation for a ratio of a probabilistic density function out of a cumulative distribution function of the standard normal distribution () () () 0, () ( ) () dg u u ug u gu gu du C u dg u du φ () + − = = +Φ− + 2 () () 0, () () u ug u gu gu C u φ () + = = +Φ 2 () () () 0 dg u ug u gu du + − = Bernoulli differential equation for a standard normal distribution () () ( ) u gu C u φ = +Φ− λ λ Φ(− ) 2 λ λ Φ(− ) λ 2λ λ − λ 0 () yu u = ( ), , 0.1, ..., 0 0.0 1.0 .9, , 1.1, ..., 2.9, 3.0 C k k λ = Φ− = 2 ( ) ( ) () () ( () max 1 1 () e ( xp 2 2 () , ) ( ) ) u u u C u z dz u z yu g u u u u z u C u u d φ φ φ π φ − −∞ −∞ Φ− > Φ− = − = = = ∴ = Φ = + Φ − ∫ ∫ ( ) u Φ− () u Φ ( ) () () () () yu u gu u u φ Φ− Φ u 0.1 k = 0.2 k = 0.3 k = 1.0 k = 0.0 k = () ( ) u u u φ = Φ− Inverse Mills ratio: 1.0 () , ( ) ( ) ( ) () If , then () 2 ( ) If , then ( ) () 2 () 2 , 0.612003, 0.2702678 u u C u u g u g φ λ λ φλ λ λ λ λ λ λ φλ λ λ φλ λ λ λ λ = =Φ− Φ− +Φ− = = = Φ− =− − = = Φ(− ) ∴ = Φ(− ) = Φ(− )= ( ), ( ), ( ), , () P h u u u v u φ ψ Φ 0.612003 λ = 2 λ λ 2λ − λ − λ − () u Φ () u φ u ( ) λ λ Φ− (1 ( )) λ λ +Φ − 2 ( ) λ λ Φ− () u Φ () u φ ( ), ( ), ( ), , () P g v v v u v φ ψ Φ () v Φ 2 () v λΦ v () () () P h u u u u φ = + Φ () () () P h u u u u φ = + Φ λ λ − 2 () () ( ) () P v g v v φ λ = Φ− +Φ 1 () () () P v g v v φ = Φ () ( ) v λ Φ − Φ− 1 () () () P v g v v φ − =− Φ Negative probability density function of truncated normal distribution at 0.612003 : () () ( ) v v λ φ ψ λ = − =− Φ− 1 () () () P v g v v φ = Φ 2 () () ( ) () P v g v v φ λ = Φ− +Φ () () ( ) v v φ ψ λ = Φ− Probability density function of truncated normal distribution at 0.612003 : () () ( ) u u λ φ ψ λ = = Φ− 2 () u λΦ 0 2λ − 1.0 − λ − λ λ λ − 0 v u = v u = v u = v u =− v u =− v u =− () (() ( )) N h u u u u φ − =− − Φ− 直角三角形による接線の傾きは，累積分布関数 の確率で，標準正規分布と逆ミルズ比の切片系 の方程式である． 共同研究者： 大西 匡光 教授 （大阪大学大学院経済学研究科 & 数理・データ科学教育研究センター） 1 1 ( () ( )) 2 2 P N h u h u u − = 1 () () ( ) N v g v v φ = Φ− 2 () () ( ) ( ) N v g v v φ λ = Φ− +Φ − () 2 ( ) φλ λ λ = Φ− () ( ) 0 ( ) () 0 () () ( ) ( ) P N P N P N P N h u h u h u h u h u h u u h u h u u − − = − − = − = − − − =− () 2 ( ) φλ λ λ = Φ− 1 () () () P v g v v φ − =− Φ v u = v u =− λ λ − 0 () 2 ( ) φλ λ λ = Φ− () 2 ( ) φλ λ λ = Φ− v u =− v u = v 2 λ λ 2λ − λ − u () (() ( )) N h u u u u φ − =− − Φ− 1 1 ( () ( )) 2 2 P N h u h u u − = λ λ − 0 () u φ 0 2λ − λ − λ () (() ( )) N h u u u u φ − =− − Φ− 1 () () () P v g v v φ − =− Φ Negative probability density function of truncated normal distribution at 0.612003 : () () ( ) v v λ φ ψ λ = − =− Φ− 0.612003 λ = λ − () u Φ () () () P h u u u u φ = + Φ 2 () () ( ) () P v g v v φ λ = Φ− +Φ v u = v u =− ( ), ( ), ( ), , () P g v v v u v φ ψ Φ () v Φ 1 () () () P v g v v φ = Φ ( ), ( ), ( ), , () P h u u u v u φ ψ Φ ( ) λ λ Φ− 2 ( ) λ λ Φ− () v u λ =Φ ( ) v u λ =Φ− ( ) v u λ = −Φ − v u λ = + ( ) 2 ( ) v u λ λ λ =Φ− + Φ− () 2 ( ) v u λ λ λ =Φ − Φ− ( ) 2 ( ) v u λ λ λ =Φ− − Φ− 1 () () () ( ) () P N vu h u h u λ λ =Φ +Φ − 2 () ( ( ) () () ( )) P N v u h u h u λ λ − =−Φ− +Φ 1 2 1 2 () () () () () () (1 2 ( ))( () ( )) 2 ( ) P N P N vu v u h u h u vu v u h u h u u u λ λ + = + − = − Φ− − = − Φ− 1 2 If , then () () 2 ( ) u v v λ λ λ λ λ λ = − = − Φ− MP4 Visual Animation PowerPoint Visual Animation
Transcript of リスクとリターンが語る円・正方形・直角三角形と...
Shingo NAKANISHI at Osaka Institute of Technology, Japan, Masamitsu OHNISHI at Osaka University, Japan.© 2019, 2-5, September, Belfast, UK, RSS (Royal Statistical Society) Annual Conference 2019© 2019, Poster Presentation and Design: Shingo NAKANISHI, Osaka Institute of Technology, Japan
Acknowledgements: The first author, Shingo NAKANISHI, would like to show my grateful to many research members at Operations Research Society of Japan, Kansai-tiku Koryukai of The Securities Analysts Association of Japan, Osaka Institute of Technology in Japan, and Osaka University in Japan.
Original References: Operations Research Society of Japan (ORSJ), Research Institute for Mathematical Sciences, Kyoto University (RIMS Kôkyûroku, 2078-10), The 15th International Symposium on Econometric Theory and Applications (SETA2019).
URL: http://www.oit.ac.jp/center/~nakanishi/english/
Geometric Characterizations and Symmetric Relations between Standard Normal Distribution and Inverse Mills Ratio based on Pythagorean Theorem
2
1
1
1
12 11
1
2
4
3
3
2
2
4
4 4
1
1
1 11
Slopes as Probabilities of standard normal distribution are connected to the right triangle by Pythagorean theorem geometrically.
1.0 x− x
Probability point: 0x = Probability point: 0.30263084( )( ( ) ( ))
( ) / ( ) 1
xx x
x x
ηφ Φφ Φ
= == −
− =
ααα
ββ
Probability point: 0.506054 ( ( ) / ( ))( ) 0.693591( ) 0.306409
( )( ) ( )
( ) / ( )
x x xx
xx xx x x
x x x x
φ ΦΦΦα Φβ Φ φ
φ Φ α β
= ==
− == −= == = = +
1
2
1
Probability point : 0( ) 1 ( ) 0.5( ) 0.5
Proportion of right triangle=1:2: 5
xx x
xΦ ΦΦ
== − − =
− =
( ) : ( ) 0.5 : 0.5 1:1x xΦ Φ− = =
Probability point : 0.67449( ) 1 ( ) 0.75( ) 0.25
Proportion of right triangle=3:4:5
xx x
xΦ ΦΦ
== − − =
− =
( ) : ( ) 0.25 : 0.75 1: 3x xΦ Φ− = =
All points whithin the circle and square are converging at one point practically because their pobability points are 0.
1t =0
1.0
1.0
1.01.0
0.0
0.0 0.00.0
Green dashed lines are probability density functions (PDF): f(x),Red dashed lines are cumulative distribution functions (CDF): F(x),Purple 1-dot chain lines are integrals of cumulative distribution functions,Red dotted lines are 1.0,Purple dotted lines are asymptotic lines whose slopes are 1.0.
(a) Type of a triangular distribution (b) Type of a uniform distribution (c) Type of an exponential distribution
(d) Type of a normal distribution
First concepts are shown as integral forms of cumulative distribution functions.
0, ( 1)1, ( 1 0)
( )1, (0 1)
0, ( 1)
xx x
f xx x
x
≤ − + − < ≤= − + < ≤ >
0, ( 0)( ) 1, (0 1)
0, ( 1)
xf x x
x
<= ≤ ≤ >
0, ( 0)( )
exp( ), ( 0)x
f xx x
<= − ≥
Aspect ratio 1.0=
F(x)F(x)F(x)
F(x)
- 40 - 20 0 20 40
- 4
- 3
- 2
- 1
0
1
t
1 2
1 2
1 , 1 ,0.5, 1.0
( 0.10, 0.12, 0.14,0.16, ,0.38, 0.40)
c cp p v
α α
α
= − = − −= = ==
0.10α =0.12α =0.14α =
- 40 - 20 0 20 40
- 6
- 4
- 2
0
2
4
t
1 2
1 2
1 , 1 ,0.5, 1.0
( 0.10, 0.12, 0.14,0.16, ,0.38, 0.40)
c cp p v
α α
α
= − = − −= = ==
0.10α =0.12α =0.14α =
If we think of the repetition game of coin tossing with a constant fee,we show you two parabolic curves of the performances both winners and losers.
We find that the maximal profit of winners is equal to its fee based on 27 percent probability.
0
( )y u
u
λ
λλ−
λ−
2λ
Interv
al of sol
id gre
en lin
e 0.7
77
≈
Regression lin
e: ( )
y uuλ=
Two dimensionalstandard normal distributionwith correlation
21 1( ) exp , ( )22
f x x xπ
= − −∞ < < ∞
( ) ( ( ) ( )) ( )Winners risk aversions
WU t t tφ λ λ λ λ λ= − Φ − = Φ −
( ) (1 ( ))LU t tλ λ= − +Φ −
* ( ) (1 ( )) ( )
Losers risk preferencesL
U t tλ λ= − +Φ − −
( )( ) its fee ( )( )W
tV t t tλ λ λ αλ
Φ −= = = =
Φ −
(1 ( ))( )(1 ( ))(1 ( ))(1 ( ))
LtV t
t
λ λλλλλ
+Φ −= −
−Φ −+Φ −
= −−Φ −
*
(1 ( ))( ) ( )(1 ( ))L
V t tλλλ
+Φ −= − −
−Φ −
((
,,
))W
EY
tασ
((
,,
))L
EY
tασ
(,
,)
WZt
ασ
(,
,)
LZt
ασ
0
1t =
0
1
tPositive ex
pected profits
0
( )Equilibrium point of inverse Mills ratio : 2( )
uφ λλ
=Φ −
0.612003λ =
Maximal loss out of ( ) is equal to its profit : λ λΦ −
Pr(
()
0)(
()|
()
0)(
)X
tE
Xt
Xt
λ<<
Φ−
2(1
)~(
,1)
UX
Nλ
λ+
=
Relations both squares and normal distribution based on 2λ
tλ
( )( )( )
uuu
ψφΦ
Inve
stor
’s p
rofit
s and
loss
es:
()
Xt
Prof
its (
()
0)X
t>
Loss
es (
()
0)X
t<
Loss
es (
()
0)X
t<
Prof
its (
()
0)X
t>
Inve
stor
’s p
rofit
s and
loss
es:
()
Xt
0
Expectation of the truncated normal distribution is 2λ−
2λ
λλ
λ
PDF of the truncatednormal distribution:
( ) , ( )( )( )
0, ( )
u uu
u
φ λλψ
λ
≤ −Φ −= > −
λ
2C
ase:
1,
(1
)~(
,1)
tX
Nλ
=+
Cas
e:
2,
(2)~
(2
,2)
tX
Nλ
=+
Cas
e:
3,
(3)~
(3
,3)
tX
Nλ
=+
Cas
e:
4,
(4)~
(4
,4)
tX
Nλ
=+
1
t
Negative expected profits
( )( )
( )
uuu
ψφ
Φ −
0
Relations both squares and normal distribution based on 2λ
Expectation of the truncated normal distribution is 2λ
2λ
λ
tλ−
λλ
λ
PDF of the truncatednormal distribution:
( ) , ( )( )( )
0, ( )
u uu
u
φ λλψ
λ
≥ −Φ −= < −
Maximal Profit out of ( ) is equal to its charge : λ λΦ −
2(
)~(
,1)
UX
tN
λλ
−=
−
2C
ase:
1,
(1
)~(
,1)
tX
Nλ
=−
Cas
e:
2,
(2)~
(2
,2)
tX
Nλ
=−
Cas
e:
3,
(3)~
(3
,3)
tX
Nλ
=−
Cas
e:
4,
(4)~
(4
,4)
tX
Nλ
=−
Pr(
()
0)(
()|
()
0)(
)X
tE
Xt
Xt
λ>>
Φ−
0
00
If we consider 1 and 1, we can estimate the equilibrium point as . The integral results are as follows.
( ) ( ( )) ( )( ) ( ( )) ( ) ( ) ( ( ) ~ ( , )),( )( ( )) ( ) (
t
x t f x t dx tx t f x t dx tX t N t t
f x t dx t f
σ µ λ
λ λ λ λλ
∞
−∞
−∞
= = = ±
− Φ −= = − +
Φ −∫∫
∫ 0
( ) ( ( ) ~ ( , )).( )( )) ( )
X t N t tx t dx t
λ λ λ λλ∞
Φ −= = −
Φ −∫
If we think of a profit as when 1, we can use ( ) ~ ( , ).Or if we think of a loss as when 1, we can use ( ) ~ ( , ).Or if we think of them as 0 when 1, we can use ( ) ~ (0, ).
t t X t N t tt t X t N t t
t X t N t
λ λλ λ
+ = +− = −
=
2 2 2 2
the interval of
If we consider the
solid green line i
correlation coefficient as Karl Pear
s nearly equal to 0son's another finding value 0.612003,
under the condition: ( ) 2 ( ) (1.777
).u y u uy u
ρ
λ λ λ+ − = −
Radius of the circle: ( 0.612003),
Correlation coefficient:.
λ
ρ λ
=
=
Relations betweenregression analysis,principal componentanalysis, and 0.612003.λ =
( ), ( ), ( ), , ( )Ph u u u v uφ ψΦ
0.612003λ =
2λλ2λ− λ−
λ−
( )uΦ
( )uφu
'' '
'
''
( ) ( ) ( ) 0( ) ( ) ( )( ) ( )( ) ( )
P P P
P
P
P
h u uh u h uh u u u uh u uh u u
φ
φ
+ − == + Φ
= Φ
=
Self-adjoint second order differential equation of a standard normal distribution:
( )λ λΦ −
(1 ( ))λ λ+Φ −
2 ( )λ λΦ −
Tangent line of ( )( ) ( ) ( ) 2 ( ) ( )( ) ( ) ( ) 2 ( ) ( )
Ph uv u u uv u u u
λ φ λ λ λ λ λλ φ λ λ λ λ λ
= Φ − + = Φ − + Φ = −= Φ + = Φ + Φ =
( )uΦ
( )uφ ( ), ( ), ( ), , ( )Pg v v v u vφ ψΦ
0. Background
We present the geometric characterizations and symmetric relations between standard normal distribution and inverse Mills ratio by circles and squares from the viewpoint with considering the height of densities such as ancient Egyptian drawing styles and using the Greek Pythagorean Theorem. First, we can clarify the integral forms of various cumulative distribution functions including standard normal distribution based on the aspect ratio (=1.0, see 1). Second, we reconsider what the several times of standard deviations multiplied by the square root of the time mean with their positive and negative expectations multiplied by the time under the condition the aspect ratio=1.0 (see 2). At this time, we can understand the following things. (1) We can find the equilibrium formulation at any real numbers of the time t (see 2). (2) Its constant number, 0.612003, was found by Karl Pearson about 100 years ago. (3) Sir David Roxbee Cox reconfirmed the value to cluster the normal distribution. (4) Truman Lee Kelley also proposed the 27 percent rule formulation. Third, we can show the parabola for maximal profits including fees. And that is equal to its fee based on the 27 percent probability. In addition to these tendencies, we can clarify the relations between winners, losers, and their banker (see 3). Fourth, we can understand two types of ordinary differential equations to explain that with circles and squares (see 4). One is the Bernoulli differential equation of inverse Mills ratio. The other is the second order linear differential equation of the integral of cumulative normal distribution function (see 1). From these tendencies, we can also get the modified intercept forms geometrically and symmetrically for maximal profits of winners, these losses of losers, and their banker’s fee. We can understand that these equations should be changing the probability points and these probabilities based on Pythagorean Theorem correctly. If the probability point is that found by Karl Pearson, we can show you that it is the special point of standard normal distribution such as Sir Cox’s proposal. Other point on the right triangle with 3:4:5 is also the third quantile of standard normal distribution. Finally, we can also realize there are many similar tendencies close to the relations between circles and squares such as Vitruvian man by Da Vinci and various Mandalas although there might not be related to normal distribution directly and historically. The ancient Egyptian drawing styles enable us to illustrate the geometric characterizations and symmetric relations between standard normal distribution and inverse Mills ratio with circle and square based on the Pythagorean theorem in the ancient Greece. We think that our ideas shall be contributed in the statistical modelling and these evaluation fields since our suggested figures should be much more easily understood and powerful than we thought.
' 2
1
2
Bernouilli differential equation of an inverse Mills ratio :
( ) ( ) ( ) 0( )( )
( )( )( ) , ( 0, inverse Mills ratio)( )
( )( ) , ( ( ))( ) ( )
P P P
P
P
P
g v ug v g vvg v
v Cvg v Cv
vg v Cv
φ
φ
φ λλ
+ + =
=Φ +
= =Φ
= = Φ −Φ − +Φ
Equiliburium point between a banker, winners and losers :
0.612003λ =( ) 2 ( )φ λ λΦ λ= −
Modified intercept form of linear equation for winners :1 1 1( ) ( )
( )Modified intercept form of linear equation for losers :
1 1 1( ) ( )( )
Modified intercept form of linear equation for t
u v
u v
φ λ φ λλ
φ λ φ λλ
− + =
Φ −
− + =
Φheir banker :
1 1 1( ) ( )( ) ( )
u vφ λ φ λλ λ
− + =
Φ − +Φ
( )vΦ 2 ( )vλΦv
( )( )
( )
Phu
uu
u
φ=
+Φ
( )( )
( )
Phu
uu
u
φ=
+Φ
λ
λ−
2( )( )
( ) ( )Pvg v
vφλ
=Φ − +Φ
1
( )
( )( )
P
v
gv
vφ=Φ
( )( )
vλ
Φ−Φ −
Negative probability density function of truncated normal distribution at 0.612003:
( )( )( )
vv
λφψ
λ
=
− = −Φ −
1
( )
( )( )
P
v
gv
vφ=Φ
2( )( )
( ) ( )Pvg v
vφλ
=Φ − +Φ
( )( )( )
vv φψλ
=Φ −
Probability density function of truncated normal distribution at 0.612003:
( )( )( )
uu
λφψ
λ
=
=Φ −
2 ( )uλΦ
02λ− 1.0− λ− λ
λ
λ−
0
vu=
vu=
vu=
vu
= −
vu
= −
vu
= −
Pythagorean Triangles show the probabilities as CDFs.Inverse Mills Ratios and PDFs are discribed as Modified Intercept Forms.
2 2
2 22 0 2
2
22
( ( , , ) )1 1 1 1 ( ( , , ) )( , , ) exp ( , , ) ( , , ) exp ( ,2 2( ( , , )) 2 2( , , )
Pr( ( , , ) 0) ( ( , , ) )1 1exp ( , , )22
WW W
WW
WW
y t t x t ty t dy t x t dxt tE Y t t tZ t
X t y t tdy t
tt
α σ α α σ αα σ α σ α σ ασ σα σ πσ πσα σ
α σ α σ αα σ
σπσ
∞ ∞
−∞
∞
−∞
+ +⋅ − ⋅ − = = =
> +−
∫ ∫
∫2
20 2
2
22
2
22
, )
1 1 ( ( , , ) )exp ( , , )22
( ( , , ) )1 1( , , ) exp ( , , )2( ( , , )) 2( , , )
Pr( ( , , ) 0) ( ( , , ) )1 1exp ( , , )22
LL L
LL
LL
t
x t t dx ttt
y t ty t dy t x
tE Y t tZ tX t y t t
dy ttt
σ
α σ α α σσπσ
α σ αα σ α σ
σα σ πσα σα σ α σ α
α σσπσ
∞
∞
−∞
∞
−∞
+−
+⋅ −
= = =< +
−
∫
∫
∫
20
22
20
22
*
0* *
1 1 ( ( , , ) )( , , ) exp ( , , )22
1 1 ( ( , , ) )exp ( , , )22
( , , ) max ( ( , , ))
( , , ) ( ( , , ))( , , ) max[ ( , , ),0]( , , ) min[
W Wt
L L
W
L
x t tt dx ttt
x t t dx ttt
U t E Y t
U t E Y tY t X tY t
τ
α σ αα σ α σσπσ
α σ α α σσπσ
α σ α σ
α σ α σα σ α σα σ
−∞
−∞
< <
+⋅ −
+−
=
===
∫
∫
( , , ),0]X tα σ
Concluding RemarksWe can understand that the slopes should be the probabilites of standard normal distribution based on the probability points.And above numbers are also the special probability points.
Intercept form of linear equation for winners :1 1 1
( )Intercept form of linear equation for losers :
1 1 1(1 ( ))1 ( )
1 ( )Intercept form of linear equation for their banker :
u v
u v
λ λ λ
λ λλλλ
− + =Φ −
− + =+Φ − +Φ −
−Φ −
−1 1 1u vλ λ
+ =( ) ( ( ) ( ))
Nh u u u uφ− = − − Φ −
( ) ( ) 0( ) ( ) 0( ) ( )( ) ( )
P N
P N
P N
P N
h u h uh u h uh u h u uh u h u u
− − =− − =− =
− − − = −
1
1
( ( )( ))
2
2
P
N
h uh u
u
−
=
λ
λ−
0
( )uφ
02λ− λ− λ( ) ( ( ) ( ))
Nh u u u uφ− = − − Φ −
1
( )
( )
( )
P
v
gv
vφ
−
= −Φ
Negative probability density function of truncated normal distribution at 0.612003:
( )( )( )
vv
λφψ
λ
=
− = −Φ −
0.612003λ =λ−
( )uΦ
( )( )
( )
Phu
uu
u
φ=
+Φ
2( )( )
( ) ( )Pvg v
vφλ
=Φ − +Φ
vu=
vu
= −
( ), ( ), ( ), , ( )Pg v v v u vφ ψΦ
( )vΦ
1
( )
( )( )
P
v
gv
vφ=Φ
( ), ( ), ( ), , ( )Ph u u u v uφ ψΦ
( )λ λΦ −
2 ( )λ λΦ −
( )v uλ= Φ
( )v uλ= Φ −
( )v uλ= −Φ −
vuλ
=+
( ) 2 ( )
vuλ
λ λ
= Φ − + Φ −( )
2(
)
v
uλ
λλ
= Φ
−Φ−
( ) 2 ( )
vuλ
λ λ
= Φ − − Φ −
1( )
( )( )
()
( )
P
N
v u
hu
hu
λ
λ
= Φ
+Φ−
2 ( ) ( ( ) ( ) ( ) ( ))P
N
v uh u
h uλ
λ
− = − Φ −+Φ
1 2
1 2
( ) ( )( ) ( )
( ) ( )(1 2 ( ))( ( ) ( ))
2 ( )
P N
P N
v u v uh u h u
v u v uh u h u
u uλλ
+= +−= − Φ − −= − Φ −
1 2
If , then( ) ( ) 2 ( )u
v vλ
λ λ λ λ λ=− = − Φ −
1
( )
( )
()
N
v
gv
vφ
= Φ−
2
()
()
()
()
N
v
gv
vφλ
=Φ−
+Φ−
( ) 2 ( )φ λ λ λ= Φ −
( ) 2 ( )φ λ λ λ= Φ −
( )2 ( )φ λλ
λ=
Φ −
2 (0)φ−
4 (0)φ−
λ λ2φ λ λ λ( ) = Φ(− )
λ λΦ(− )
λ
λ
(1 )λ λ+ Φ(− )
(1 )λ λ−Φ(− )1.02λ
2CDF of (0,1 ) : ( )N uΦ2PDF of (0,1 ) : ( )N uφ
0.0
PDF of truncated normal distribution:( ) , ( )( )( )
0, ( )
u uu
u
φ λλψ
λ
≤ −Φ −= > −
( )( )( )
uuu
ψφΦ
1 ( )1 ( )
λλλ
+ Φ −−Φ −
λ λΦ(− )
λ λ(1−Φ(− ))
1 3 =0.6120046191 2 3
λ +≅
+
Our concept of the integration of standard normal disribution with a square and a circle is similar to “Mandala” and “Vitruvian Man” which is one of da Vinci’s works coincidentally. The original reference websites are https://en.wikipedia.org/wiki/Mandala, https://ja.wikipedia.org/wiki/曼荼羅 , https://en.wikipedia.org/wiki/Vitruvian_Man (Access date: July, 3rd, 2017).Moreover, we are able to refer the idea and figure of Prof. Ida’s website which is discribed that the ratio should be less than the inverse number of the golden ratio. Reference: Ida, T., “Vitruvian Man by Leonardo da Vinci and the Golden Ratio” , http://www.crl.nitech.ac.jp/~ida/education/VitruvianMan/index.html (Access date: July, 3rd, 2017).
We estimate the practical approximated value such as “Squaring the circle” which is simmilar to a problem proposed by ancient geometers. https://en.wikipedia.org/wiki/Squaring_the_circle (Access date: November 18th, 2017)
u
1. First concept as integrals of various cumulative distributions
2. Second concept as distances of probability points based on the time t. There should be maximal losses against the positive returns or maximal profits against the negative returns by the 27 percent probabilities.
3. Relations between winners, losers, and their banker by the equilibrium formulation and
4. Main concept as the relations between Pythagorean theorem, differential equations, circles, and squares on standard normal distribution
( ) 0.2702678λΦ − =0.612003tα λσ
= =
5. Special case as the geometric characterizationsand rotationally symmetric relations between winners, losers, and their banker based on the condition: and
MP4Visual
Animation
PowerPointVisual
Animation
( ) 0.2702678.λΦ − =0.612003λ =
That is, the fee is equal to .α λ
3:4:5
著者・作成者： 中西 真悟 （大阪工業大学情報センター・准教授）
共同研究者 ： 大西 匡光 （大阪大学大学院経済学研究科 & 数理・データ科学教育研究センター・教授）
謝 辞： ご助言，ご支援，ご協力を賜った多くの日本オペレーションズ・リサーチ学会員の皆様，
日本証券アナリスト協会関西地区交流会の皆様， 大阪大学と大阪工業大学の教職員の皆様に厚く御礼を述べます．
また，若い頃の恩師 中易秀敏 先生，栗山仙之助 先生と，リスクと確率論について手解きしてくれた
若い頃の上司 亀島鉱二 先生，志垣一郎 先生に感謝いたします．
掲示図の出典： 日本オペレーションズ・リサーチ学会論文と研究発表会資料 (ORSJ)， 京都大学数理解析研究所講究録 (RIMS Kôkyûroku, 2078-10) と修正版 , 第 15 回計量経済学の理論と応用に関する国際シンポジウムへの投稿論文とプレゼンテーション (SETA2019)， 2019 年王立統計学会年次大会ポスター・プレゼンテーション (RSS2019).
リスクとリターンが語る円・正方形・直角三角形と標準正規分布の対称的な幾何学的特性 URL: http://www.oit.ac.jp/center/~nakanishi/
2019 年 9 月 13 日に
大阪工業大学梅田キャンパスで開催の
イノベーションデイズにおいて
展示した作品をベースに作成
Version 2
注 1： 左側の白枠中は，大阪工業大学イノベーションデイズでの掲載作品です．一部は OR 学会 2019 年秋季研究発表会や 2019 年王立統計学会国際会議でも発表しています．
右側は 2019 年の王立統計学会主催の国際会議ポスタープレゼンテーションで発表した掲載作品です．円や曲線を含むポスターのデザインは大阪の夜空に打ち上がる「花火の色彩と背景」を表現しています．
注 2： ポスターの背景のデザインの色合いは本学のスクールカラー「紺青，大阪工大ブルー」，コミュニケーションマークのカラー「シアン」，
淀川や大阪湾も含めて世界へ発信できる「水の都 大阪」をイメージして作成しています． © 中西真悟（大阪工業大学），大西匡光（大阪大学），2019
勝者，敗者，胴元の均衡点:
0.612003λ =( ) 2 ( )φ λ λΦ λ= −
( ), ( ), ( ), , ( )Ph u u u v uφ ψΦ
vu=
vu
= −
From 大阪工業大学大宮キャンパス南東側：生駒山 北西側：六甲山
1
1
2
1
1
1
12 11
1
2
4
3
3
2
2
4
4 4
1
11
Probabilities of standard normal distribution are connected to the right triangle by Pythagorean theorm geometrically.
1
Probability point : 0( ) 1 ( ) 0.5( ) 0.5
Proportion of right triangle=1:2: 5
xx x
xΦ ΦΦ
== − − =
− =
( ) : ( ) 0.5 : 0.5 1:1x xΦ Φ− = =
Probability point : 0.67449( ) 1 ( ) 0.75( ) 0.25
Proportion of right triangle=3:4:5
xx x
xΦ ΦΦ
== − − =
− =
( ) : ( ) 0.25 : 0.75 1: 3x xΦ Φ− = =
All points whithin the circle and square are converging at one point practically because their pobability points are 0.
勝者，敗者，胴元の均衡点:
0.612003λ =( ) 2 ( )φ λ λΦ λ= −
標準正規分布に二階線形微分方程式とベルヌーイ型微分方程式を用いた．古代エジプトの絵画図の技法のように奥行きを取り除いて回転させて重ね合わせてみると，ピタゴラスの定理とパラメトリック方程式による円，正方形，直角三角形の美しい幾何学模様が描ける．
緑色は標準正規分布による二階線形微分方程式に関連する曲線
赤色は標準正規分布（逆ミルズ比）によるベルヌーイ型微分方程式に関連する曲線
水色は標準正規分布による二階線形微分方程式を用いたパラメトリック方程式に関連する曲線
1 1 1( ) ( )( )
1 1 1( ) ( )( )
1 1 1( ) ( )( ) ( )
k kk
k kk
k kk k
φ φ
φ φ
φ φ
− + =
Φ −
− + =
Φ
− + =
Φ − +Φ
勝者の切片系の方程式：
敗者の切片系の方程式：
胴元の切片系の方程式：
大阪の街から見える生駒山と六甲山の山の高さはどことなく縦横比が同じときの標準正規分布の高さに似ている．同時に，歴史を通じた文化を感じながら曼荼羅のような幾何学模様を描ける．
λ λ λ λ
(0)
φ
()
λλ
Φ−
2(
)λ
λΦ
−
()
λλ
Φ−
λ
1.02λ
( )uφ0.0
1.0
( )( ) 2 ( )
f uuλ
λ λ
= −Φ − + Φ −
( )(1
())
2(
)
f u
uλ
λλ
= −−Φ
−+
Φ−
2 ( )( )
()
hu
uu
u
φ=
−Φ−
( )( )( )
uu φψλ
=Φ −
u
Tangential equation :
2 ( ) ( )dh u udu
= −Φ −
Probability density function of a trancated normal distribution :
Probability density function of a standard normal distribution :Cumulative distribution function of a standard normal distribution :
( )uφ( )uΦ −
φ λ λ λ( ) = 2 Φ(− )
Equiliburium point between a banker, winners and losers :
0.612003λ =
Intercept form of a linear equation for winners :
Intercept form of a linear equation for losers :
Intercept form of a linear equation for a banker :
2
2
2
1 1 1
1 1 1(1 )1
11 1 1
u
u
u
φλ λ λ
φλ λλλ
λ
φλ λ
− + =Φ(− )
− + =+ Φ(− ) + Φ(− )
−Φ(− )
− + =
2
2
( )(
( )
)( )u
f uh u
uφ
φψ
22
2
( ) ( )d h u udu
φ=
( ) 2( )φ λ λ
λ=
Φ −
Equiliburium point of an inverse Mills ratio :
Utility function for winners :( ) ( ( ) ( )) ( )WU t t tφ λ λ λ λ λ= − Φ − = Φ −
( ) ( )1 '' '
2 2 2'1 ' 1
2 2
Self-adjoint differential equation: ( ) ( ( ) ( ) ( )) 0( ) ( ) ( ) ( ) 0u h u uh u h uu h u u h u
φφ φ
−
− −
+ − =− =
Variable coefficient second order linear homogeneous differential equation for a standard normal distribution :
22 2
22
2
2
( ) ( ) ( ) 0,1(0) ( (0)),2
(0) 1 ( (0))2
d h u dh uu h udu du
h
dhdu
φπ
+ − =
= =
= − = −Φ
(1 ( )) 2 ( ) ( 2 0)( )
( ) 2 ( ) (0 2 )u u
f uu uλ λ λ λ
λ λ λ λ− −Φ − + Φ − ≤ ≤
= −Φ − + Φ ≤ ≤
2 uλΦ( )
uΦ( )
2
2
Cumulative distribution f
Probabilistic density funct
unction of a standard norma
ion of a standard normal distribution:
1 1( ) exp2
l distribution: 1 1( ) exp
221or ( ) e2
2
u
u
u u du
u
u
φ
π
π
π
−∞
Φ = −
Φ
− =
= −
∫21xp
2u
u du−
−∞
− ∫
22π
12π
uφλ
( )Φ(− )
λ− λ0
λ λΦ(− )2λ λ φ λΦ(− ) = ( )
u
2λ
λ
1.0 uΦ( )
uλ2 Φ( )
( )u
uφλ( )
Φ(− ) + Φ − ( )u
uφλ( )
Φ(− ) + Φ
( )uu
φ( )Φ ( )
uu
φ( )Φ −
uλ2 Φ(− )
uΦ(− )
max( ) ( ) 2 ( )
max( ) ( ) 2 ( )
uu
uu
φ φ λ λλ λφ φ λ λλ λ
( ) ( )= =
Φ − + Φ − Φ −( ) ( )
= =Φ − + Φ Φ −
If thenC λ= Φ(− )
λλ
φ λ λ λ
= 0.612003Φ(− ) = 0.2702678
( ) = 2 Φ(− )
0.302630840.3810856
ηη
φ η η
=Φ(− ) =
( ) = Φ(− )
Inverse Mills ratio known as hazard function:
Survival ratio of a standard normal distribution:
( )f u u=( )f u u= −
Probabilistic density function of a trancated normal distribution:
( ,1.0)η
2
Bernoulli differential equation for a ratioof a probabilistic density functionout of a cumulative distribution functionof the standard normal distribution
( ) ( ) ( ) 0, ( )( )
( )
dg u uug u g u g udu C u
dg udu
φ( )+ − = =
+ Φ −
+ 2( ) ( ) 0, ( )( )
uug u g u g uC uφ( )
+ = =+ Φ
2( ) ( ) ( ) 0dg u ug u g udu
+ − =
Bernoulli differential equation for a standard normal distribution
( )( )( )
ug uC u
φ=
+ Φ −
λ λΦ(− )2λ λΦ(− )
λ
2λ
λ− λ0
( )y u u=
( ), , 0.1, ..., 00.0 1.0.9, , 1.1, ..., 2.9, 3.0C k kλ= Φ − =
2
( )
( ) ( )
( ) (
( ) max
1 1( ) e
(
xp22
( ),
)
()
)
u
u
u C
u z dz
u z
y u g
u u
u uz
uC u
u
d
φ
φ
φπ
φ
−
−∞
−∞
Φ − >
Φ − = − =
= = ∴
=
Φ
=+ Φ
−
∫∫
( )uΦ − ( )uΦ
( )( )
( )
( )( )
y u
u
g u
uu
φΦ −Φ
u
0.1k =
0.2k =0.3k =
1.0k =
0.0k =
( )( )uuu
φ=Φ −
Inverse Mills ratio:
1.0
( ) , ( )( ) ( )
( )If , then ( )2 ( )
If , then ( ) ( ) 2( ) 2 , 0.612003, 0.2702678
uu Cu
u g
u g
φ λλ
φ λλ λ λλ
λ λ φ λ λ λφ λ λ λ λ λ
= = Φ −Φ − +Φ −
= = =Φ −
= − − = = Φ(− )∴ = Φ(− ) = Φ(− ) =
( ), ( ), ( ), , ( )Ph u u u v uφ ψΦ
0.612003λ =
2λλ2λ− λ−
λ−
( )uΦ
( )uφu
( )λ λΦ −
(1 ( ))λ λ+Φ −
2 ( )λ λΦ −
( )uΦ
( )uφ ( ), ( ), ( ), , ( )Pg v v v u vφ ψΦ
( )vΦ 2 ( )vλΦv
( )( )
( )
Phu
uu
u
φ=
+Φ
( )( )
( )
Phu
uu
u
φ=
+Φ
λ
λ−
2( )( )
( ) ( )Pvg v
vφλ
=Φ − +Φ
1
( )
( )( )
P
v
gv
vφ=Φ
( )( )
vλ
Φ−Φ −1
( )
( )
( )
P
v
gv
vφ
−
= −Φ
Negative probability density function of truncated normal distribution at 0.612003:
( )( )( )
vv
λφψ
λ
=
− = −Φ −
1
( )
( )( )
P
v
gv
vφ=Φ
2( )( )
( ) ( )Pvg v
vφλ
=Φ − +Φ
( )( )( )
vv φψλ
=Φ −
Probability density function of truncated normal distribution at 0.612003:
( )( )( )
uu
λφψ
λ
=
=Φ −
2 ( )uλΦ
02λ− 1.0− λ− λ
λ
λ−
0
vu=
vu=
vu=
vu
= −
vu
= −
vu
= −
( ) ( ( ) ( ))Nh u u u uφ− = − − Φ −
直角三角形による接線の傾きは，累積分布関数の確率で，標準正規分布と逆ミルズ比の切片系の方程式である．
共同研究者： 大西 匡光 教授 （大阪大学大学院経済学研究科 & 数理・データ科学教育研究センター）
1
1
( ( )( ))
2
2
P
N
h uh u
u
−
=
1
( )
( )
()
N
v
gv
vφ
= Φ−
2
()
()
()
()
N
v
gv
vφλ
=Φ−
+Φ−
( ) 2 ( )φ λ λ λ= Φ −
( ) ( ) 0( ) ( ) 0( ) ( )( ) ( )
P N
P N
P N
P N
h u h uh u h uh u h u uh u h u u
− − =− − =− =
− − − = −
( ) 2 ( )φ λ λ λ= Φ −
1
( )
( )
( )
P
v
gv
vφ
−
= −Φ
vu=
vu
= −
λ
λ−
0
( ) 2 ( )φ λ λ λ= Φ −
( ) 2 ( )φ λ λ λ= Φ −
vu
= −vu=
v
2λλ2λ− λ−u
( ) ( ( ) ( ))Nh u u u uφ− = − − Φ −
1
1
( ( )( ))
2
2
P
N
h uh u
u
−
= λ
λ−
0
( )uφ
02λ− λ− λ( ) ( ( ) ( ))
Nh u u u uφ− = − − Φ −
1
( )
( )
( )
P
v
gv
vφ
−
= −Φ
Negative probability density function of truncated normal distribution at 0.612003:
( )( )( )
vv
λφψ
λ
=
− = −Φ −
0.612003λ =λ−
( )uΦ
( )( )
( )
Phu
uu
u
φ=
+Φ
2( )( )
( ) ( )Pvg v
vφλ
=Φ − +Φ
vu=
vu
= −
( ), ( ), ( ), , ( )Pg v v v u vφ ψΦ
( )vΦ
1
( )
( )( )
P
v
gv
vφ=Φ
( ), ( ), ( ), , ( )Ph u u u v uφ ψΦ
( )λ λΦ −
2 ( )λ λΦ −
( )v uλ= Φ
( )v uλ= Φ −
( )v uλ= −Φ −
vuλ
=+
( ) 2 ( )
vuλ
λ λ
= Φ − + Φ −( )
2(
)
v
uλ
λλ
= Φ
−Φ−
( ) 2 ( )
vuλ
λ λ
= Φ − − Φ −
1( )
( )( )
()
( )
P
N
v u
hu
hu
λ
λ
= Φ
+Φ−
2 ( ) ( ( ) ( ) ( ) ( ))P
N
v uh u
h uλ
λ
− = − Φ −+Φ
1 2
1 2
( ) ( )( ) ( )
( ) ( )(1 2 ( ))( ( ) ( ))
2 ( )
P N
P N
v u v uh u h u
v u v uh u h u
u uλλ
+= +−= − Φ − −= − Φ −
1 2
If , then( ) ( ) 2 ( )u
v vλ
λ λ λ λ λ=− = − Φ −
MP4Visual
Animation
PowerPointVisual
Animation
