Root Example

In[]:=
$Assumptions=s>0&&σ>0&&N>0&&d>0;
In[]:=
CorrelationMean[s_,σ_,N_]:=s^2+σ^2/N(*EquationB.5*)
In[]:=
CorrelationVariance[s_,σ_,N_]:=s^2*σ^2*(1+3/N)+(N+1)σ^4/N^2(*EquationB.23*)
In[]:=
CorrelationDPrime[s_,σ_,N_]:=(CorrelationMean[s,σ,N]-CorrelationMean[0,σ,N])/Sqrt[(CorrelationVariance[s,σ,N]+CorrelationVariance[0,σ,N])/2]
In[]:=
CorrelationDPrimeSimple=Simplify[CorrelationDPrime[s,σ,N]](*B.29*)
Out[]=
2
N
2
s
N(3+N)
2
s
2
σ
+2(1+N)
4
σ
In[]:=
CorrelationThreshold=Solve[{CorrelationDPrime[s,σ,N]==d,s>0},s]
Out[]=
{{sRoot[-2
2
d
4
σ
-2
2
d
N
4
σ
+(-3
2
d
N
2
σ
-
2
d
2
N
2
σ
)
2
#1
+2
2
N
4
#1
&,2]}}
In[]:=
CorrelationThresholdExp=ToRadicals[CorrelationThreshold]
Out[]=
s-
1
2
2
d
2
σ
+
3
2
d
2
σ
N
-
2
d
2
N
(16+9
2
d
+16N+6
2
d
N+
2
d
2
N
)
4
σ
2
N

In[]:=
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