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

limit factor analysis

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

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

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

factor normal distribution (Faktor Normalverteilung)

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=1,01

factor normal distribution simulation (Faktor Normalverteilung Simulation)

scattering probability limit factor normal distribution sigma zeta phi deisenroth

σd(ϕd;n) = σd^(±ζd(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)

ϕd(σd;n)=ϕd(σd;n=1)^-n

ζd(ϕd;n) = ±√(-ln(ϕd)/n)

probability limit factor sigma deisenroth

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

probability limit exponent zeta deisenroth

ζd(ϕd;n) = ±√(-ln(ϕd)/n)

probability limit factor phi deisenroth

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

probability limit factor sigma deisenroth σd(ϕd;n)

limit factor distribution deisenroth

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

probability phi deisenroth ϕd

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^0,) = 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) = ±√(-ln(ϕd))

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