ϕ'(x/µ) = abs(2n*e^(-n*(ln(x/µ)/ln(σd))²)*ln(x)/((ln(σd))^2/x)
hypothesis:
ϕ(x/µ) = e^(-n*((ln(x/µ))/(ln(σd))²
σ(ϕ) = x(ϕ;n)/µ = σd^±√(-ln(ϕ)/n) = x/µ
x/µ = σd^±√(-ln(ϕ)/n)
x = µ*σd^±√(-ln(ϕ)/n)
x(ϕ;σd;µ;n) = µ*σd^±√(-ln(ϕ)/n)
probability density normal distribution phi'(x/µ) = ϕ'(x/µ)
ϕ'(x/µ) = abs(2n*e^(-n*(ln(x/µ)/ln(σd))²)*ln(x)/((ln(σd))^2/x)
test:
monte carlo simulation excel N = 10000
excel random generator probability factor normal distribution sigma zeta phi deisenroth
x(ϕ)/µ = ?
x(ϕ)/µ = σd^±√(-ln(ϕ)
= σd^(((GANZZAHL((EXP(LN((0,5+ZUFALLSZAHL())))))-0,5)*2)*((-LN(ZUFALLSZAHL()))^0,5)^(1^0,5))
ϕ(x/µ) = probability factor phi deisenroth
probatility factor sigma deisenroth
σ(ϕ) = probability factor sigma deisenroth = x(ϕ)/µ
probability factor normal distribution sigma zeta phi deisenroth
σ(ϕ;n) = σd^(±ζ(ϕ)/n^0,5) = x(ϕ;n)/µ
σ(ϕ;n)= σd^(±√((-ln(ϕ)/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)
ζ(ϕ) = ±√(-ln(ϕ))
ζ(ϕ;n) = ±√(-ln(ϕ)/n)
true arithmetic mean (µari) and true geometric mean (µgeo or µ)
µari = true arithmetic mean (wahrer arithmetischer mittelwert)
µgeo or µ = 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)
standard factor sigma deisenroth
σd(µari/µ) = e^(2∗ln(µari/µ))^0¸5
sd(xgeo/xari) = e^(2∗ln(xari/xgeo))^0¸5
standard normal factor sigma deisenroth
σd(µari/µ) = e^(2∗ln(µari/µ))^0¸5
sd(xgeo/xari) = e^(2∗ln(xari/xgeo))^0¸5
standard probability factor sigma deisenroth
σd(µari/µ) = e^(2∗ln(µari/µ))^0¸5
sd(xgeo/xari) = e^(2∗ln(xari/xgeo))^0¸5
standard limit factor sigma deisenroth
σd(µari/µ) = e^(2∗ln(µari/µ))^0¸5
sd(xgeo/xari) = e^(2∗ln(xari/xgeo))^0¸5
standard probability limit factor sigma deisenroth
σd(µari/µ) = e^(2∗ln(µari/µ))^0¸5
sd(xgeo/xari) = e^(2∗ln(xari/xgeo))^0¸5
standard probability limit exponent zeta deisenroth
ζ(ϕ=0,368) = 1
standard probability limit factor phi deisenroth
ϕ(ζ=1) = e^-ζ² = e^-1² = 0,368
standard sigma deisenroth
σd(µari/µ) = e^(2∗ln(µari/µ))^0¸5
standard zeta deisenroth
ζ(ϕ=0,368) = 1
standard phi deisenroth
ϕ(ζ=1) = e^-ζ² = e^-1² = 0,368
confidence limit base deisenroth
σd(µari/µ) = e^(2∗ln(µari/µ))^0¸5
confidence limit factor sigma deisenroth
σd(ϕ;n) = σ^(±ζ(ϕ)/n^0,5)
confidence limit exponent zeta deisenroth
ζ(ϕ) = ±√(-ln(ϕ))
confidence limit factor phi deisenroth
ϕ(σ;n) = e^(-n*((ln(x(ϕ;n)/µ))/(ln(σd))²)
standard confidence limit factor sigma deisenroth
σd(µari/µ) = e^(2∗ln(µari/µ))^0¸5
standard confidence limit exponent zeta deisenroth
ζ(ϕ=0,368) = 1
standard confidence limit factor phi deisenroth
ϕ(ζ=1) = e^-ζ² = e^-1² = 0,368
development sigma zeta phi score
for
true standard probability factor sigma deisenroth σd
true confidence limit factor sigma deisenroth σ(ϕ)
true confidence limit exponent zeta deisenroth ζ(ϕ)
true probability phi deisenroth ϕ(x)