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

limit factor normal distribution

probability limit factor normal distribution sigma zeta phi deisenroth = σd(ϕd;n) = σd^±ζd(ϕd;n) = σd^(±√(-ln(ϕd)/n)) = x(ϕd;n)/µgeo ; σd=2

standard normal limit factor sigma deisenroth

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

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

limit factor normal distribution = phi(x/µ;sigma;n) = ϕ(x/µ;σ;n) = limit factor distribution

ϕ(x/µ;σ;n) = e^(-n*(ln(x/µ)/ln(σ))²)

grenzfaktor normalverteilung = phi(x/µ;sigma;n) = ϕ(x/µ;σ;n) = grenzfaktorverteilung

ϕ(x/µ;σ;n) = e^(-n*(ln(x/µ)/ln(σ))²)

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

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

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

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

phi Grenzfaktor Normalverteilung = ϕ(x/µ;σ;n) = e^(-n*(ln(x/µ)/ln(σ))²)

limit factor normal distribution 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))

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)

ϕd(ζd) = e^ζd(ϕd;n) = e^(-ln(ϕd)/n)

ϕd(n) = ϕd^-n

probability limit factor sigma deisenroth

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

probability limit factor phi deisenroth

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

ϕd(n) = ϕd^-n

probability limit exponent zeta deisenroth

ζd(ϕd;n) = ±√(-ln(ϕd)/n)

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

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

scattering probability limit factor normal distribution sigma zeta phi deisenroth

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

σd(ϕd;n) = σd^(±ζd(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) = ±√(-ln(ϕd)/n)

probability limit base sigma deisenroth