Project Definition ϕ(x/µ)=?

project definition phi(x) = ϕ(x-µ) or ϕ(x/µ) ? 

https://sites.google.com/site/scatteringfactordistribution/

question:  

ϕ(x/µ)=? 

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 = σ(x/µ) = σd^±√(-ln(ϕ)/n) 

x = µ*σd^±√(-ln(ϕ)/n) 

σ(ϕ) = x(ϕ;n)/µ = x(ϕ;σd;µ;n)  = 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))

ϕ(x/µ) =  probability factor phi deisenroth 

probability 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(σ))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)

probability limit base sigma deisenroth 

standard probability limit base sigma deisenroth 

standard probability limit factor sigma deisenroth 

σ(µari/µ) =  e^(2∗ln(µari/µ))^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 

σ(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 

σ(µari/µ) =  e^(2∗ln(µari/µ))^0¸5

confidence limit factor sigma deisenroth 

σ(ϕ;n) = σ^(±ζ(ϕ)/n^0,5)

confidence limit exponent zeta deisenroth

ζ(ϕ) = ±√(-ln(ϕ))