Geometric Characterizations of Standard Normal Distribution

About the title

“Geometric Characterizations of Standard Normal Distribution

- Two Types of Differential Equations, Relationships with Square and Circle, and Their Similar Characterizations -”,

we will modify some things as follows to contribute mathematics and sciences precisely on November 14, 2018.

Important modification:

Present modified version: ℎ2(𝑢𝑢) =ℎ𝑃𝑃(𝑢𝑢),

𝜙𝜙(𝑢𝑢)⁻¹(ℎ_𝑃𝑃^′′(𝑢𝑢)+𝑢𝑢ℎ_𝑃𝑃^′(𝑢𝑢)− ℎ_𝑃𝑃(𝑢𝑢)) = 0.

Previous misspelled version: ℎ2(𝑢𝑢) =𝑚𝑚𝑃𝑃(𝑢𝑢),

𝜙𝜙(𝑢𝑢)�𝑚𝑚_𝑃𝑃^′′(𝑢𝑢)−𝑢𝑢𝑚𝑚_𝑃𝑃^′(𝑢𝑢)− 𝑚𝑚_𝑃𝑃(𝑢𝑢)�= 0.

The other equation in this paper: 𝑔𝑔(𝑢𝑢) =𝑔𝑔𝑃𝑃(𝑢𝑢),

𝑔𝑔_𝑃𝑃^′(𝑢𝑢) +𝑢𝑢𝑔𝑔_𝑃𝑃(𝑢𝑢) +𝑔𝑔(𝑢𝑢)²= 0.

If we think of three equations as follows

ℎ_𝑃𝑃(𝑢𝑢) =𝜙𝜙(𝑢𝑢) +𝑢𝑢𝑢𝑢(𝑢𝑢), 𝑚𝑚𝑃𝑃(𝑢𝑢) =𝑢𝑢(𝑢𝑢)

𝜙𝜙(𝑢𝑢), 𝑔𝑔𝑃𝑃(𝑢𝑢) =𝜙𝜙(𝑢𝑢)

𝑢𝑢(𝑢𝑢), these differential equations are expressed as

𝜙𝜙(𝑢𝑢)⁻¹(ℎ_𝑃𝑃^′′(𝑢𝑢) +𝑢𝑢ℎ_𝑃𝑃^′(𝑢𝑢)− ℎ_𝑃𝑃(𝑢𝑢)) = 0, 𝜙𝜙(𝑢𝑢)�𝑚𝑚_𝑃𝑃^′′(𝑢𝑢)− 𝑢𝑢𝑚𝑚_𝑃𝑃^′(𝑢𝑢)− 𝑚𝑚_𝑃𝑃(𝑢𝑢)�= 0, 𝑔𝑔_𝑃𝑃^′(𝑢𝑢) +𝑢𝑢𝑔𝑔_𝑃𝑃(𝑢𝑢) +𝑔𝑔(𝑢𝑢)²= 0.

The upper two of the equations are self-adjoint differential equations.

And we find that there is the following relationship between above three equations 𝑔𝑔𝑃𝑃(𝑢𝑢), ℎ𝑃𝑃(𝑢𝑢) and 𝑚𝑚𝑃𝑃(𝑢𝑢). That is,

−𝑔𝑔𝑃𝑃′(𝑢𝑢) 𝑔𝑔_𝑃𝑃(𝑢𝑢) =

ℎ𝑃𝑃(𝑢𝑢) ℎ_𝑃𝑃^′(𝑢𝑢) =

𝑚𝑚𝑃𝑃′(𝑢𝑢) 𝑚𝑚_𝑃𝑃(𝑢𝑢).

© Shingo NAKANISHI, Osaka Institute of Technology, November 14, 2018.

I have a plan to speak about them on November 22, 2018 to be modified.

λ λ λ λ

(0)φ

()λλΦ−2()λλΦ−()λλΦ−λ 1.0

2λ

2 ( )λΦ −u ( )u Φ −

( )u φ 0.0

1.0

( ) ( ) 2 ( ) f u

λ u

λ λ

= −Φ −

+ Φ − ( ) (1

( ))

2 (

) f u

λ u λ

λ

= − − Φ−

+ Φ

−

2( ) ( )

( ) h u

u u u

=φ

− Φ

− ( ) ( )

( ) u φ u ψ ⁼Φ −λ

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 :φ( )u

( )u Φ −

φ λ( ) = 2 Φ(− )λ λ 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

λ λ λ φ

λ λ φ

λ λ

λ λ λφ

− + =

Φ(− )

− + =

+ Φ(− )

 + Φ(− )

 − Φ(− )

 

− + =

2 2( ) (

( ) ( )u) h uf u

u φ ψφ

2 2 2

( ) ( ) d h u u

du =φ

( ) 2

( )

φ λ λ

λ ⁼ Φ − Equiliburium point of an inverse Mills ratio :

Utility function for winners :

( ) ( ( ) ( )) ( )

U tW = φ λ − Φ −λ λ t= Φ −λ λ t

( ) ( )

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,

(0) 1 ( (0)), (0) 1 (2 (0))

2 d h u udh u h u

du du

h dh

du

π φ

+ − =

= =

= − = −Φ

(1 ( )) 2 ( ) ( 2 0)

( ) ( ) 2 ( ) (0 2 )

u u

f u u u

λ λ λ λ

λ λ λ λ

− − Φ − + Φ − ≤ ≤

= −Φ − + Φ ≤ ≤

Modified Version of our study,

“Geometric Characterizations of Standard Normal Distribution - Two Types of Differential Equations,

Relationships with Square and Circle, and Their Similar Characterizations -” ,

http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2078-10.pdf

© Shingo NAKANISHI, Osaka Institute of Technology, November 4, 2018.