PROOF OF MY LEAST WHOLE NORMAL FUNCTION

THE LWN FUNCTION

If S is the frequency population of population N and µ is the frequency mean of mean ʘ at population S and N respectively. The time in which µ is occurring is given as τ. The function of the parameters is given as;

ɭ(τ)=ατ-τ^0.5-β  where α and β are denoted as quantile coefficient of τ and quantile constant respectively.

PROOF!

The expected mean of the population S is E_s=ʘµτ and its variance is given as V_s=(ʘs²τ+µ²σ²τ)  where s² is variance of the population S and σ² is variance of population N.

At normal approximation; the probability at which the occurring of S is greater than ϒ% is given as;

P(S>-s)=P[S>-(s-ʘµτ)/(ʘs²τ+µ²σ²τ)^0.5]=ϒ%

This implies that;

-s+ʘµτ=ф^-1(ϒ%)(ʘs²+µ²σ²)^0.5_*τ^0.5

-s/ф^-1(ϒ%)(ʘs²+µ²σ²)^0.5+ʘµτ/ф^-1(ϒ%)(ʘs²+µ²σ²)^0.5=τ^0.5

But let α=ʘµ/ф^-1(ϒ%)(ʘs²+µ²σ²)^0.5

And β=s/ф^-1(ϒ%)(ʘs²+µ2σ²)^0.5

Therefore ατ-τ^0.5-β=0,  

Where α and β are quantile coefficient of τ and quantile constant respectively.

Proving the equation quadratically , we have

Τ^0.5=[1+(1+4αβ)^0.5]/2α

 The symbols S, N, ʘ, µ, and τ have different meanings, based on the area of application of the LWN Function.





REFERENCE
Adongo Ayine William(Me), Diary(2009), Posted(EMS-Bolga Branch) to Ghana Academy of Arts and Sciences in the Year 2010.