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

probability limit phi(x/µ) = ϕ(x/µ)

probability limit phi(x/µ) = ϕ(x/µ) = e^(-n*((ln(x/µ))/(ln(σd))²) = phi(sigma) = ϕ(σ)

sigma(phi) limit factor distribution deisenroth = σ(ϕ;n) = σd^(±ζ(ϕ;n)) = σd^±√(-ln(ϕ)/n) = x/µ = sigma factor deisenroth = σ

sigma(phi) limit factor distribution deisenroth = σ(ϕ;n) = σd^(±ζ(ϕ;n)) = σd^±√(-ln(ϕ)/n)

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;n) = σd^±√(-ln(ϕd)/n) = x(ϕd;n)/µgeo

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

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

ζd(ϕd;n) = ±√(-ln(ϕd)/n)

probability limit factor sigma deisenroth

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

probability limit factor phi deisenroth

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

probability limit exponent zeta deisenroth

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)

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

xari = estimated arithmetic mean (geschätzter arithmetischer mittelwert)

xgeo = estimeted geometric mean (geschätzter geometrischer mittelwert)

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

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

probability limit base sigma deisenroth

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

probability limit factor sigma deisenroth

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) = ±√(-ln(ϕd))

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;n=1) = 1

standard probability limit factor phi deisenroth

ϕd(ζd=1;n=1) = e^-(ζd²/n) = e^-1² = 0,368

standard sigma deisenroth

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

standard zeta deisenroth

ζd(ϕd=0,368;n=1) = 1

standard phi deisenroth

ϕd(ζd=1;n=1) = e^-(ζd²/n) = 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;n=1) = 1

standard confidence limit factor phi deisenroth

ϕd(ζd=1;n=1) = e^-(ζd²/n) = e^-1² = 0,368

scattering factor (Streufaktor)

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 factor (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(36,8%;n) = σd^(1/n^0,5)

confidence limit exponent zeta deisenroth

ζdij = ln(xgeoj/xgeoi)/(ln(σdi)/√ni + ln(σdj)/√nj) = true confidence limit exponent zeta deisenroth

Zdij=(ln(xgeoj/xgeoi)/(ln(sdi)/√(ni -2)+ln(sdj)/√(nj-2))^(1/2^0,5) = estimated confidence limit exponent zeta deisenroth

xgeoi = estimated geomean i

σdi = true standard scattering factor simga deisenroth i

sdi = estimated standard scattering factor simga deisenroth i

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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