phi cumulative factor distribution function = ϕc = 0,5*ϕ(x/µ;σd;n)

phi cumulative factor product distribution function deisenroth = ϕcp(xgeo/µgeo*sd>1) = ϕcp^n

cumulative probability limit factor product distribution function phi-cum-product deisenroth = ϕcp(σcp;n) and sigma-cum-product deisenroth = σcp(ϕcp;n)

phi-cum deisenroth

ϕc(x;n)=1/2*ϕ(x;n)=1/2∗e^((-(ln⁡(x/µ)/(ln⁡σd))^2 ) )^n

phi-cum-product deisenroth

ϕcp(x;n)=(1/2 *ϕ(x;n))^n=(1/2∗e^(-(ln⁡(x/µ)/(ln⁡σd))^2 ) )^n

sigma-cum-product deisenroth

σcp(ϕ;n) = x/µ (ϕ;n)=σd^(±√(-lnϕ)/n))

phi deisenroth ϕ or ϕd

ϕ(x;n) = ϕ(x)^n = (e^(-(ln⁡(x/µ)/(ln⁡σd))^2 ))^n

sigma deisenroth = σ = x/µ  =σ(σd;ϕ;n) = x/µ = σd^(±√(-ln(ϕ)/n ))

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

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

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

xari(xi;n) = 1/n*(x1+x2+ ....+xn)

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 

σ(ϕ;n)= σd^(±√((-ln(ϕd)/n))) = x(ϕ;n)/µ

σ(ϕ;n) = σd^(±ζd(ϕd;n) = x(ϕ;n)/µ

ϕ(x;n) = e^(-n*((ln(x/µ))/(ln(σd))²) 

x(ϕ;n) = µ∗σd^(±√((-ln(ϕ)/n)))

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

 ζd(ϕ;n) = ±√(-ln(ϕd)/n) 

factor normal distribution (Faktor Normalverteilung)