Anhang    

Die tanφ-Formel

[100]         1 / tan2φ     =  ( σ²_x – σ²_y ) / σ_xy ;                     Voraussetzung:     σ²_δ = σ²_ε , ( λ = 1)

[101]                                =  ( σ²_ξ + σ²_δ – σ²_η – σ²_ε ) /σ_xy 

[102]         1 / tan2φ   =  σ²_ξσ²_η / σ_ξη ;         ß = tanφ ≥ ß_e

alternativ

[103]    unter σ²_δ = σ²_{ε  ;}           (σ²_x – σ_xy/ß) = (σ²_y – ßσ_xy) | *ß ,     quadratische Gleichung für ß

Der Achsenabschnitt unter  LEGÉDRE / GALTON

[104]      ( α = µY – ß_ξηµX )   ≥  (α = µY – ß_eµX )

Rettungsversuche

für GALTON

[105]     ∂e²/∂ß = 0     -->   ß  =  xy/x²  =  y/x  ≠  ß    -->    µß  =  σ_xy /σ²_x  ≠  σ_y/σ_x  ≠  ß

für GAUSS 

[106]   ∂ε²/∂ß = 0     -->    ß  =  xy/(x - ∂)²  = xy/ξ² ≠ ß     aber   µß = σ_xy/σ²_ξ = ß 

für AUTOR

[107]   ∂0²/∂ß = 0    -->    ß  =  ξη/ξ² = η/ξ =  ß    -->  µß = σ_ξη/σ²_ξ = signσ_ξη σ_η/σ_ξ

Die ultimativen ß-pur-Formeln

                                                               ß-pur = F(α)

Sätze      [108]    α_y =  µY  -  ß_yµX

                                α_x =  µX  -  ß_xµY

                 [109]                         ß_y   =  σ_xy/ (σ²_x - σ²_δ)

                                                 ß_x  =  σ_xy / (σ²_y - σ²_ε)

[110]              ß_y ß_x =  σ_xy² / (σ²_x - σ²_δ) (σ²_y - σ²_ε) = σ²_ξ σ²_η / σ²_ξ σ²_η = 1

α = 0

[111]                   0_α =  µY - ß_yµX  | : µX

[112]                  ß_y =  µY / µX,         ß_x = analog = µX / µY

                     ß_y ß_x  =  1

α ≠ 0

[113]                                                   ß_y =  α_y / α_x | * α_x

[114]            ß_y (µX  -  ß_xµY)                =  µX  -  ß_xµY

[115]            ß_y µX – (ß_y ß_x | =1) µY   =  µY   -  ß_yµX

[116]                                       2 ß_y µX  = 2µY | : (2/ µX)

                                                  ß_y        =  µY / µX,          ß_x = analog = µX / µY

Deterministische Simulation

[117]   unter          µX_ξ = µX   | X = X_ξ + δ,      µ_δ = 0,  

                                µY_η = µY   | Y = Y_η  + ε,      µ_ε = 0

Tabelle 1     deterministische Simulation    α_y = 0

[118]                                         Y_η =  X_ξ (ß|=2)

                                                   ß_y  =  µY_η / µX_ξ  = 2

Tabelle 2      deterministische Simulation     α_y ≠ 0

[119]                                        Y_η  =  X_ξ [ß|=(-3)]

                                                 ß_y  =  µY_η / µX_ξ  = (-3)

 Tabelle 3      Simulation mit abhängigen Fehlern

 [120]