phi factor normal distribution ϕ(x/µ;σd;n) = e^(-n*(ln(x/µ)/ln(σd))²)

probability exponent = ζ(ϕ;n) = ±√(-ln(ϕ)/n)

probability exponent zeta(phi;n)

ζ(ϕ;n) = ±√(-ln(ϕ)/n) = probability exponent zeta deisenroth = ζd(ϕd;n) = ln(x/µ)/ln(σd)/√n

probability exponent ζ(ϕ;n) = ±√(-ln(ϕ)/n)

probability factor phi(x/µ;sigma;n) deisenroth = ϕ(x/µ;σ;n)

ϕ(x/µ;σ;n) = e^(-n*(ln(x/µ)/ln(σ))²)

probability factor sigma(phi;n) deisenroth = σ(ϕ;n)

σ(ϕ;n) = σ^±√(-ln(ϕ)/n) = x(ϕ;n)/µ

x(ϕ;n) = µ∗σ^±√(-ln(ϕ)/n)

x(phi,sigma,µ,n) deisenroth

x(ϕ;σ;µ;n) = µ∗σ^±√(-ln(ϕ)/n)

x(zeta,sigma,µ) deisenroth

x(ζ;σ;µ) = µ∗σ^±ζ

x(zeta,sigma,phi;µ) deisenroth

x(ζ;σ;ϕ;µ) = µ∗σ^±ζ(ϕ)

x(zeta,sigma,µ,n) deisenroth

x(ζ;σ;µ;n) = µ∗σ^(±ζ/√n)

probability exponent zeta deisenroth = ζ = ζd

ζ = ±√(-ln(ϕ)) = ln(x/µ)/ln(σ)

probability exponent zeta(phi) deisenroth = ζ(ϕ)

ζ(ϕ) = ±√(-ln(ϕ)) = ln(x/µ)/ln(σ) = probability exponent zeta(phi) deisenroth

probability exponent zeta(phi;n) deisenroth = ζ(ϕ;n)

ζ(ϕ;n) = ±√(-ln(ϕ)/n) = ln(x/µ)/ln(σ)/√n= probability exponent zeta(phi;n) deisenroth

probability exponent zeta(x/µ;sigma) deisenroth = ζ(x/µ;σ)

ζ(x/µ;σ) = ln(x/µ)/ln(σ) = ±√(-ln(ϕ)) = probability exponent zeta(x/µ;sigma) deisenroth

probability exponent zeta(x/µ;sigma;n) deisenroth = ζ(x/µ;σ;n)

ζ(x/µ;σ;n) = ln(x/µ)/ln(σ)/√n = ±√(-ln(ϕ)/n) = probability exponent zeta(x/µ;sigma;n) deisenroth

probability factor phi(zeta) deisenroth = ϕ(ζ)

ϕ(ζ) = e^-(ζ^2)

probability factor phi(zeta,n) deisenroth = ϕ(ζ;n)

ϕ(ζ;n) = e^-(ζ^2/n) = e^-(ζ^2/n )

probability factor phi(n) deisenroth = ϕ(n)

ϕ(n) = ϕ^(-1/n) = ϕ^n

standard factor sigma deisenroth σ and s (standardfaktor sigma deisenroth)

σ = e^(2*ln(µari/µgeo))^0,5 = true standard factor sigma (wahrer standardfaktor sigma deisenroth) = σd

s = e^(2*ln(xari/xgeo))^0,5 = estimated standard factor sigma (geschätzter standardfaktor sigma deisenroth) = sd

probability limit factor sigma deisenroth

probability limit factor function phi deisenroth

factor normal distribution deisenroth

normal distribution deisenroth

normal distribution deisenroth and monte carlo simulation

confidence limit factor sigma deisenroth

confidence limit exponent zeta deisenroth

Project Definition

question:

σxgeo(ϕd;n)=?

hypothesis:

σd(ϕd;n) = x(ϕd;n)/µgeo = σd^(±√((-ln(ϕd)/n)))

σxgeo(ϕd;n) = xgeo(ϕd;n)/µgeo = σd^-(((√-ln(ϕd))*-2*ϕd)/n^0,5)

test:

monte carlo simulation excel xgeo(n)*sd(n); (N = 10000)

excel random generator scattering probability limit factor normal distribution sigma zeta phi deisenroth

x(ϕd)/µgeo = ?

x(ϕd)/µgeo = σd^(±√((-ln(ϕd))))

= σd^(((GANZZAHL((EXP(LN((0,5+ZUFALLSZAHL())))))-0,5)*2)*((-LN(ZUFALLSZAHL()))^0,5)^(1^0,5))

scattering probability limit factor normal distribution sigma zeta phi deisenroth

σd(ϕd;n) = σd^(±ζd(ϕd)) = x(ϕd;n)/µgeo

σd(ϕd;n)= σd^(±√((-ln(ϕd)/n))) = x(ϕd;n)/µgeo

ϕd(x;n) = e^(-n*((ln(x/µgeo))/(ln(σd))²)

x(ϕd;n) = µgeo∗σd^(±√((-ln(ϕd)/n)))

ϕd(σd;n) = e^(-n*((ln(x(ϕd;n)/µgeo))/(ln(σd))²)

ζd(ϕd;n) = ±√(-ln(ϕd)/n)

true arithmetic mean (µari) and true geometric mean (µgeo)

µari = true arithmetic mean (wahrer arithmetischer mittelwert)

µgeo = true geometric mean (wahrer geometrischer mittelwert)

µari = µgeo*e^(0,5*ln(σd)^2)

µgeo = µari/e^(0,5*ln(σd)^2)

estimated arithmetic mean (xari) and estimated geometric mean (xgeo)

xari = estimated arithmetic mean (geschätzter arithmetischer mittelwert)

xgeo = estimeted geometric mean (geschätzter geometrischer mittelwert)

xgeo(x_i;n) = e^(1/n*((ln(x₁)+ln(x₂)+.....+ln(x_n)))

xgeo(sd;xari) = xari/e^(0,5*ln(sd)^2)

xari(x_i;n) = 1/n*(x₁+x₂+ ....+x_n)

xari(sd;xgeo) = xgeo*e^(0,5*ln(sd)^2)

true standard scattering probability limit factor sigma deisenroth σd and estimated standard scattering probability limit factor sigma deisenroth sd

σd = true standard scattering probability limit factor sigma deisenroth (wahrer standardstreufaktor deisenroth)

sd = estimated standard scattering probability limit factor sigma deisenroth (geschätzter standardstreufaktor deisenroth)

σd = e(2*ln(µari/µgeo))^0,5

sd = e(2*ln(xari/xgeo))^0,5

probability limit base sigma deisenroth σd

σd(µari/µgeo) = e^(2∗ln(µari/µgeo))^0¸5

probability limit factor sigma deisenroth σd

σd(ϕd;n) = σd^(±ζd(ϕd)/n^0,5) = x(ϕd;n)/µgeo

σd(ϕd;n=1)= σd^(±√((-ln(ϕd)/n=1))) = x(ϕd;n)/µgeo

x(ϕd;n=1) = µgeo∗σd^(±√((-ln(ϕd)/(n=1))))

x(ϕd;n>1) = µgeo∗σd^(±√((-ln(ϕd)/(n=1))))

probability limit exponent zeta deisenroth

ζd(ϕd;n) = ±√(-ln(ϕd)/n)

probablity limit factor phi deisenroth

ϕd(σd;n) = e^(-n*((ln(x(ϕd;n)/µgeo))/(ln(σd))²)

standard probability limit base sigma deisenroth

standard probability limit factor sigma deisenroth

σd(µari/µgeo) = e^(2∗ln(µari/µgeo))^0¸5

standard probability limit exponent zeta deisenroth

ζd(ϕd=0,368) = 1

standard probability limit factor phi deisenroth

ϕd(ζd=1) = e^-ζd² = e^-1² = 0,368

standard sigma deisenroth

σd(µari/µgeo) = e^(2∗ln(µari/µgeo))^0¸5

standard zeta deisenroth

ζd(ϕd=0,368) = 1

standard phi deisenroth

ϕd(ζd=1) = e^-ζd² = e^-1² = 0,368

confidence limit base deisenroth

σd(µari/µgeo) = e^(2∗ln(µari/µgeo))^0¸5

confidence limit factor sigma deisenroth

σd(ϕd;n) = σd^(±ζd(ϕd)/n^0,5)

confidence limit exponent zeta deisenroth

ζd(ϕd;n) = ±√(-ln(ϕd)/n)

σd(µari/µgeo) = e^(2∗ln(µari/µgeo))^0¸5

confidence limit factor phi deisenroth

ϕd(σd;n) = e^(-n*((ln(x(ϕd;n)/µgeo))/(ln(σd))²)

standard confidence limit factor sigma deisenroth

σd(µari/µgeo) = e^(2∗ln(µari/µgeo))^0¸5

standard confidence limit exponent zeta deisenroth

ζd(ϕd=0,368) = 1

standard confidence limit factor phi deisenroth

ϕd(ζd=1) = e^-ζd² = e^-1² = 0,368

scattering factors (Streufaktoren)

σd = e^(2∗ln(µari/µgeo))^0¸5 = true standard scattering factor sigma deisenroth (wahrer Standardstreufaktor)

sd = e^(2∗ln(xari/xgeo))^0¸5 = estimated standard scattering factor sigma deisenroth (geschätzter Standardstreufaktor)

σari = µari/µgeo = e^(0,5∗(ln σd)^² ) = true µari scattering factor (wahrer Aristreufaktor)

sari = xari/xgeo = e^(0,5∗(ln sd)^²) = estimated xari scattering factor (geschätzter Aristreufaktor)

σp = σd(ϕd;n)*x/µgeo = true scattering factor product (wahres Streufaktorprodukt)

sp = sd*xgeo/µgeo = estimated scattering factor product (geschätztes Streufaktorprodukt)

σgeo = µgeo/µgeo = 1 = true µgeo scattering factor (wahrer µgeo Streufaktor)

sgeo = xgeo/xgeo = 1 = estimated xgeo scattering factor (geschätzter xgeo Streufaktor)

σxgeo = xgeo/µgeo = true xgeo scattering factor (wahrer xgeo Streufaktor)

σµgeo(n,%) = µgeo(n,%)/µgeo(100%) = true µgeo scattering factor (wahrer µgeo Streufaktor)

σxgeo(n,%) = xgeo(n,%)/xgeo(100%) = true xgeo scattering factor (wahrer xgeo Streufaktor)

σd(n,%) =x(n,%)/µgeo =σd^(ζd(%)/n^0,5)= true x confidence limit factor sigma deisenroth (wahrer µgeo Vertrauensgrenzfaktor)

σd(n,%)=µgeo(n,%)/µgeo=σd^(ζd(%)/n^0,5)=true µgeo confidence limit factor sigma deisenroth (wahre µgeo Vertrauensgrenze)

CLd(n,%,σd) = σd^(ζd(%)/n^0,5) * x = true x confidence limit deisenroth (wahre x Vertrauensgrenze)

sd(n,%) = xgeo(n,%,sd)/xgeo = estimated xgeo confidence limit factor deisenroth (gesch. VGF)

CLd(n,%,sd) = estimated xgeo confidence limit deisenroth (geschätzte Vertrauensgrenze)

extreme limit factors (extreme Grenzfaktoren)

sd(n,100%) = sd^±(ζd(100%)/n^0,5) = sd^(±0/n^0,5) = 1 = xgeo(n,100%)/xgeo(100%) = 1

sd(n,0%) = sd^±(ζd(0%)/n^0,5) = sd^±(∞/n^0,5) = ∞/1 and 1/∞ = ∞/xgeo(100%) and xgeo(100%)/∞=0

sd(n,100%) = 1

sd(n,0%) = 0 and ∞

σd(n,100%) = 1

σd(n,0%) = 0 and ∞

σd(n,36,8%) = σd^(1/n^0,5)

confidence limit exponent zeta deisenroth

ζdij = ln(xgeo_j/xgeo_i)/(ln(σd_i)/√n_i+ ln(σd_j)/√n_j) = true confidence limit exponent zeta deisenroth ij

Zdij=?= estimated confidence limit exponent zeta deisenroth ij

xgeo_i = estimated geomean i

σd_i = true standard scattering factor simga deisenroth i

sd_i = estimated standard scattering factor simga deisenroth i

probability phi deisenroth ϕd(ζd)

ϕd(ζd) = e^-ζd² = true probability phi deisenroth

Pd(Zd) = e^-Zd² = estimated probability phi deisenroth

probability limit phi deisenroth ϕdij(ζdij)

ϕdij(ζdij) = e^-ζdij² = true probability limit phi deisenroth

Pdij(Zdij) = e^-Zdij² = estimated probability limit phi deisenroth

probability exponent zeta deisenroth ζd(ϕd)

ζd(ϕd) = ±√(-ln(ϕd)) = true probablitiy exponent zeta deisenroth

Zd(Pd) = ±√(-ln(Pd))= estimated probablitiy exponent zeta deisenroth

probability limit exponent zeta deisenroth ζdij(ϕdij)

ζdij(ϕdij) = ±√(-ln(ϕdij))= true probablitiy limit exponent zeta deisenroth ij

Zdij(Pdij) = ±√(-ln(Pdij))= estimated probablitiy limit exponent zeta deisenroth ij

probability limit factor sigma deisenroth σdij

σdij = σd_i^(ζdij/ni^0,5) = true probability limit factor sigma deisenroth ij

sdij = ? = estimated probability limit factor sigma deisenroth ij

probability limit factor sigma deisenroth σdi(ζdi;ni)

σdi(ζdi;ni) = σd_i^(ζdi/ni^0,5) = true probability limit factor sigma deisenroth i

sdi(Zdi,ni) = ? = estimated probability limit factor sigma deisenroth i

probability limit factor sigma deisenroth σd(%;n)

σd(%;n) = σd^(ζd(%)/n^0,5) = x(%;n)/µgeo = true xgeo probability limit factor sigma deisenroth

sd(%;n) = ? = xgeo(%;n)/µgeo = estimated xgeo probability limit factor sigma deisenroth

standard probability limit factor sigma deisenroth σd

σd = e^(2∗ln(µari/µgeo))^0¸5 = true standard probability limit factor sigma deisenroth

sd = e^(2∗ln(xari/xgeo))^0¸5 = estimated standard probability limit factor sigma deisenroth

Impressum

Frank Deisenroth

Seligenstädter Weg 4

63796 Kahl am Main

Tel.: 0179 / 110 3096

deisenroth@gmail.com

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