Mean Deviation

Author

Eric W. Weisstein
November 23, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/MeanDeviation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MeanDeviation.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Code

(*MeanDeviationC[dist_,opts___?OptionQ]:=Module[{m=Mean[dist],x,i=DistributionDomain[dist],x1,x2},
{x1,x2}=If[Head[i]===Interval,i[[1]],{-1,1}∞];
Integrate[PDF[dist,x] Abs[x-m],{x,x1,x2},opts]
]*)
MeanDeviationC[dist_]:=Module[{x,m=Mean[dist]},ExpectedValue[Abs[x-m],dist,x]]

MeanDeviationD[dist_]:=Module[{m=Mean[dist],k},
Sum[PDF[dist,k] Abs[k-m],{k,∞}]
]

Can't do this because domains are not included correctly in built-in distributions

Print[{#[[1]],{mad=TimeConstrained[FullSimplify[MeanDeviationD[#[[1]],Assumptions->#[[2]]],#[[2]]],300],
mad//N}//Timing
}]&/@
{
{BetaDistribution},
{CauchyDistribution[a,b],a∈Reals&&b>0},
{ChiDistribution[ν],ν>0},
(*{ExponentialDistribution[λ],λ>0},*)
{ExtremeValueDistribution[α,β],α∈Reals&&β>0},
(*{GammaDistribution[α,β],α>0&&β>0},*)
{GumbelDistribution[α,β],α>0&&β>0},
{HalfNormalDistribution[θ],θ>0},
{LaplaceDistribution[α,β],α∈Reals&&β>0},
{LogisticDistribution[μ,β],μ∈Reals&&σ>β},
{LogNormalDistribution[μ,σ],μ∈Reals&&σ>0},
{MaxwellDistribution[σ],σ>0},
{NormalDistribution[μ,σ],μ∈Reals&&σ>0},
(*{RayleighDistribution[σ],σ>0},*)
{TriangularDistribution[{a,b}],a<b},
{TriangularDistribution[{a,b},c],a<c<b},
{UniformDistribution[{a,b}],a<b},
{WeibullDistribution[α,β],α>0&&β>0}
};

BetaDistribution:((2 α (Beta3[(α/(α+β)), α, β])-2 (α+β) (Beta3[(α/(α+β)), α+1, β]))/((α+β) (Beta[α, β])))

pdf=PDF[dist=BetaDistribution[α,β],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[x-m],{x,0,1},assum=Assumptions->α>0&&β>0],assum]

CauchyDistribution: undef

pdf=PDF[dist=CauchyDistribution[a,b],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[x-m],{x,-∞,∞},assum=Assumptions->a∈Reals&&b>0],assum]

ChiDistribution:((2 (Sqrt[2]) ((Gamma[(ν/2)]) (Gamma2[((ν+1)/2), (((Gamma[((ν+1)/2)])^2)/((Gamma[(ν/2)])^2))])-(Gamma[((ν+1)/2)]) (Gamma2[(ν/2), (((Gamma[((ν+1)/2)])^2)/((Gamma[(ν/2)])^2))])))/((Gamma[(ν/2)])^2))

pdf=PDF[dist=ChiDistribution[ν],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[x-m],{x,0,∞},assum=Assumptions->ν>0],assum]

ChiSquareDistribution:(((((2 e))^(-ν/2)) (ν^(ν/2+1)) (ν+2))/(Gamma[(ν/2)+2]))

pdf=PDF[dist=ChiSquareDistribution[ν],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[x-m],{x,0,∞},assum=Assumptions->ν>0],assum]

ExponentialDistribution:(2/(e λ))

With[{d=ExponentialDistribution[λ]},ExpectedValue[Abs[x-Mean[d]],d,x]]

MeanDeviationC[ExponentialDistribution[λ],Assumptions->λ>0]

pdf=PDF[dist=ExponentialDistribution[λ],x]

m=Mean[dist]

Integrate[pdf Abs[x-m],{x,0,∞},Assumptions->λ>0]

ExtremeValueDistribution:

pdf=PDF[dist=ExtremeValueDistribution[α,β],x]

m=Mean[dist]

Integrate[pdf Abs[x-m],{x,-∞,∞},Assumptions->α∈Reals&&α>0]

FRatioDistribution:

pdf=PDF[dist=FRatioDistribution[𝓃,𝓂],x]

m=Simplify[Mean[dist],𝓂>2]

Integrate[pdf Abs[x-m],{x,0,∞},Assumptions->𝓃>0&&𝓂>2]

FullSimplify[%,𝓃>0&&𝓂>2]

GammaDistribution:((2 (e^(-α)) (α^α) β)/(Gamma[α]))

pdf=PDF[dist=GammaDistribution[α,β],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,0,∞},Assumptions->(assum=α>0&&β>0)],assum]

GumbelDistribution:β ((Subscript[G, 2,1]^(0,2))(e,(1/ℽ)|({{1,1}, {0}}))-(ExpIntegralEi[sinh(ℽ)-cosh(ℽ)]))

pdf=PDF[dist=GumbelDistribution[α,β],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,-∞,∞},Assumptions->(assum=α∈Reals&&β>0)],assum]

HalfNormalDistribution:((2 (erf((1/(Sqrt[π])))+(e^(-1/π))-1))/θ)

pdf=PDF[dist=HalfNormalDistribution[θ],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,0,∞},Assumptions->(assum=θ>0)],assum]

InverseGammaDistribution:((2 (e^(1-α)) (((α-1))^(α-2)) β)/(Gamma[α]))

pdf=PDF[dist=InverseGammaDistribution[α,β],x]

m=Simplify[Mean[dist],α>1]

FullSimplify[Integrate[pdf Abs[m-x],{x,0,∞},Assumptions->(assum=α>1&&β>0)],assum]

InverseGaussianDistribution:

pdf=PDF[dist=InverseGaussianDistribution[μ,λ],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,0,∞},Assumptions->(assum=μ>0&&λ>0)],assum]

LaplaceDistribution:β

pdf=PDF[dist=LaplaceDistribution[μ,β],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,-∞,∞},Assumptions->(assum=μ∈Reals&&β>0)],assum]

LevyDistribution: undef

pdf=PDF[dist=LevyDistribution[μ,σ],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,k,∞},Assumptions->(assum=k>0&&α>0)],assum]

LogisticDistribution:2 β log(2)

pdf=PDF[dist=LogisticDistribution[μ,β],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,-∞,∞},Assumptions->(assum=μ∈Reals&&β>0)],assum]

LogNormalDistribution:2 erf((σ/(2 (Sqrt[2])))) (e^(μ+((σ^2)/2)))

pdf=PDF[dist=LogNormalDistribution[μ,σ],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,0,∞},Assumptions->(assum=μ∈Reals&&σ>0)],assum]

MaxwellDistribution:-4 (e^(-4/π)) (Sqrt[(2/π)]) ((e^(4/π)) erfc((2/(Sqrt[π])))-1) σ

pdf=PDF[dist=MaxwellDistribution[σ],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,0,∞},Assumptions->(assum=σ>0)],assum]

NormalDistribution:(Sqrt[(2/π)]) σ

pdf=PDF[dist=NormalDistribution[μ,σ],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,-∞,∞},Assumptions->(assum=μ∈Reals&&σ>0)],assum]

ParetoDistribution:(α/(Sqrt[2 π]))

pdf=PDF[dist=NormalDistribution[k,α],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,k,∞},Assumptions->(assum=k>0&&α>0)],assum]

RayleighDistribution:(Sqrt[2 π]) erfc(((Sqrt[π])/2)) σ

pdf=PDF[dist=RayleighDistribution[σ],x]

m=Mean[dist]

Integrate[pdf Abs[x-m],{x,0,∞},Assumptions->σ>0]

StudentTDistribution:((Sqrt[ν])/((ν-1) (Beta[(ν/2), (1/2)])))

pdf=PDF[dist=StudentTDistribution[ν],x]

m=Simplify[Mean[dist],ν>1]

FullSimplify[Integrate[pdf Abs[m-x],{x,0,∞},Assumptions->(assum=ν>1)],assum]

StudentTDistribution:((2 (Sqrt[ν]) σ-(ν^(ν/2)) (σ^ν) ((((μ^2)+ν (σ^2)))^((1/2)-(ν/2))))/((ν-1) (Beta[(ν/2), (1/2)])))

pdf=PDF[dist=StudentTDistribution[μ,σ,ν],x]

m=Simplify[Mean[dist],ν>1]

FullSimplify[Integrate[pdf Abs[m-x],{x,0,∞},Assumptions->(assum=ν>0&&μ∈Reals&&σ>0)],assum]

FullSimplify[Integrate[pdf Abs[m-x],{x,0,∞},Assumptions->(assum=ν>1&&μ>0&&σ>0)],assum]

TriangularDistribution:((b-a)/6)

pdf=PDF[dist=TriangularDistribution[{a,b}],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,-∞,∞},Assumptions->(assum=a<b)],assum]

TriangularDistribution:(Piecewise[{{((2 (((-2 a+b+c))^3))/(81 (a-b) (a-c))), a+b<2 c}, {((2 (((a-2 b+c))^3))/(81 (a-b) (b-c))), a+b>2 c}}])

pdf=PDF[dist=TriangularDistribution[{a,b},c],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,-∞,∞},Assumptions->(assum=a<c<b)],assum]

UniformDistribution:((b-a)/4)

pdf=PDF[dist=UniformDistribution[{a,b}],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,-∞,∞},Assumptions->(assum=a<b)],assum]

WeibullDistribution:

pdf=PDF[dist=WeibullDistribution[α,β],x]

m=Mean[dist]

FullSimplify[Integrate[pdf Abs[m-x],{x,0,∞},Assumptions->(assum=α>0&&β>0)],assum]

Discrete

BernoulliDistribution:-2 (p-1) p

?BernoulliDistribution

PDF[BernoulliDistribution[p],n]

Mean[BernoulliDistribution[p]]

s=FullSimplify[Sum[PDF[BernoulliDistribution[p],n]Abs[n-p],{n,0,1}],0<p<1]

MeanDeviation[RandomInteger[BernoulliDistribution[.2],10^6]//N]

-2 (-1+p) p/.{p->.2}

BetaBinomialDistribution

pdf=PDF[dist=BetaBinomialDistribution[α,β,N],n]

m=Mean[dist]

Sum[-((Beta[n+α,-n+N+β] Binomial[N,n])/(Beta[α,β]))(n-((N α)/(α+β))),{n,0,Floor[((N α)/(α+β))]}]+Sum[((Beta[n+α,-n+N+β] Binomial[N,n])/(Beta[α,β]))(n-((N α)/(α+β))),{n,Floor[((N α)/(α+β))]+1,∞}]

FullSimplify[%,N>0&&N∈Integers&&α>0&&β>0]

BetaNegativeBinomialDistribution

pdf=PDF[dist=BetaNegativeBinomialDistribution[α,β,N],n]

m=Mean[dist]

Sum[-((Beta[N+α,n+β] Binomial[-1+n+N,-1+N])/(Beta[α,β]))(n-((N α)/(α-1))),{n,0,Floor[((N α)/(α-1))]}]+Sum[((Beta[N+α,n+β] Binomial[-1+n+N,-1+N])/(Beta[α,β]))(n-((N α)/(α-1))),{n,Floor[((N α)/(α-1))]+1,∞}]

FullSimplify[%,N>0&&N∈Integers&&α>0&&β>0]

BinomialDistribution: 2 (((1-p))^(N-(Floor[N p]))) (p^((Floor[N p])+1)) ((Floor[N p])+1) (Binomial[N, (Floor[N p])+1])

?BinomialDistribution

pdf=PDF[BinomialDistribution[N,p],n]

m=Mean[BinomialDistribution[N,p]]

Sum[pdf Abs[n-m],{n,0,N}]

Sum[-(((1-p))^(-n+N)) (p^n) (n-N p) Binomial[N,n],{n,0,Floor[N p]}]+Sum[(((1-p))^(-n+N)) (p^n) (n-N p) Binomial[N,n],{n,Floor[N p]+1,N}]

FullSimplify[%,0<p<1&&N>0]

p==1/2

%/.p->1/2

p==1/2, N odd

FullSimplify[(2^(-N)) Binomial[N,1+Floor[(N/2)]] (1+Floor[(N/2)]),Mod[N,2]==1&&N>0]

FullSimplify[FunctionExpand[(2^(-1-N)) (1+N) Binomial[N,((1+N)/2)]-((N!!)/(2(N-1)!!))],Mod[N,2]==1]

p==1/2, N even

FullSimplify[(2^(-N)) Binomial[N,1+Floor[(N/2)]] (1+Floor[(N/2)]),Mod[N,2]==0]

FullSimplify[FunctionExpand[(2^(-1-N)) (2+N) Binomial[N,1+(N/2)]-(((N-1)!!)/(2(N-2)!!))],Mod[N,2]==0]

Check

MeanDeviation[RandomInteger[BinomialDistribution[10,.2],10^6]//N]

2 (((1-p))^(N-Floor[N p])) (p^(1+Floor[N p])) Binomial[N,1+Floor[N p]] (1+Floor[N p])/.{p->.2,N->10}

Trott

With[{m=Floor[n p]},((1-p)^(-2-m+n)*p^(1+m)*Gamma[1+n]*(-2*(-1+p)*(1+m-n*p)*Gamma[3+m]*Gamma[-1-m+n]*Hypergeometric2F1[1,1+m-n,2+m,p/(-1+p)]+2*p*Gamma[2+m]*Gamma[-m+n]*Hypergeometric2F1[2,2+m-n,3+m,p/(-1+p)]))/(Gamma[2+m]*Gamma[3+m]*Gamma[-1-m+n]*Gamma[-m+n])]/.{p->.2,n->10}

Overly complicated Trott form

(M. Trott, pers. comm., June 28, 2002)

(∑_(r=0)^n)\[LeftBracketingBar]n p-r\[RightBracketingBar] (({{n}, {r}})) (p^r) (((1-p))^(n-r))==((((1-p))^(-m+n-2)) (p^(m+1)) Γ(n+1) (2 p Γ(m+2) Γ(n-m) (Subscript[, 2])(Subscript[F, 1])(2,m-n+2;m+3;(p/(p-1)))-2 (p-1) (m-n p+1) Γ(m+3) Γ(-m+n-1) (Subscript[, 2])(Subscript[F, 1])(1,m-n+1;m+2;(p/(p-1)))))/(Γ(m+2) Γ(m+3) Γ(-m+n-1) Γ(n-m))

m=⌊n p⌋

Sum[Abs[n p-r] Binomial[n,r] p^r (1-p)^(n-r),{r,0,n}]==((1-p)^(-2-m+n)*p^(1+m)*Gamma[1+n]*(-2*(-1+p)*(1+m-n*p)*Gamma[3+m]*Gamma[-1-m+n]*Hypergeometric2F1[1,1+m-n,2+m,p/(-1+p)]+2*p*Gamma[2+m]*Gamma[-m+n]*Hypergeometric2F1[2,2+m-n,3+m,p/(-1+p)]))/(Gamma[2+m]*Gamma[3+m]*Gamma[-1-m+n]*Gamma[-m+n])

FullSimplify[Table[Sum[Abs[n p-r] Binomial[n,r] p^r (1-p)^(n-r),{r,0,n}],{n,3}],0≤p≤1]//ColumnForm

With[{p=1/2},Table[Sum[Abs[k-n/2]Binomial[n,k] p^n,{k,0,n}],{n,10}]]

With[{p=1/2},Table[Sum[Abs[n p-r] Binomial[n,r] p^r (1-p)^(n-r),{r,0,n}],{n,10}]]

FullSimplify[(Sum[Binomial[n,k](m-k),{k,0,m-1}]+Sum[Binomial[n,k](k-m),{k,m+1,n}])/2^n/.m->n/2]

Table[%18,{n,2,10,2}]//FullSimplify

FullSimplify[(Sum[Binomial[n,k](m-k-1/2),{k,0,m}]+Sum[Binomial[n,k](k-m+1/2),{k,m+1,n}])/2^n/.m->(n-1)/2]

Table[%23,{n,1,15,2}]//FullSimplify

Table[If[OddQ[n],n!!/2/(n-1)!!,(n-1)!!/2/(n-2)!!],{n,15}]

Table[With[{p=1/2},
m=Floor[n p];((1-p)^(-2-m+n)*p^(1+m)*Gamma[1+n]*(-2*(-1+p)*(1+m-n*p)*Gamma[3+m]*Gamma[-1-m+n]*Hypergeometric2F1[1,1+m-n,2+m,p/(-1+p)]+2*p*Gamma[2+m]*Gamma[-m+n]*Hypergeometric2F1[2,2+m-n,3+m,p/(-1+p)]))/(Gamma[2+m]*Gamma[3+m]*Gamma[-1-m+n]*Gamma[-m+n])],
{n,10}]

DiscreteUniformDistribution:(Piecewise[{{(N/4), (Mod[N, 2])==0}, {(((N-1) (N+1))/(4 N)), (Mod[N, 2])==1}}])

PDF[DiscreteUniformDistribution[{1,N}],k]

Mean[DiscreteUniformDistribution[{1,N}]]

s=Sum[(1/N)Abs[n-((1+N)/2)],{n,N}]

s2=FullSimplify[%,N∈Integers]

FullSimplify[s2,Mod[N,2]==0]

FullSimplify[s2,Mod[N,2]==1]

Piecewise[{{N/4,Mod[N,2]==0},{(N-1)(N+1)/(4N),Mod[N,2]==1}}]

GeometricDistribution:2 (((1-p))^(Floor[1/p])) (Floor[(1/p)])

?GeometricDistribution

pdf=PDF[GeometricDistribution[p],n]

m=Mean[GeometricDistribution[p]]

s=Sum[pdf Abs[n-m],{n,0,∞}]

Solve[1+n-1/p==0,n]

Sum[p(((1-p))^n)Abs[1+n-(1/p)],{n,0,∞}]

Sum[-p(((1-p))^n)(1+n-(1/p)),{n,0,Floor[((1-p)/p)]}]+Sum[p(((1-p))^n)(1+n-(1/p)),{n,Floor[((1-p)/p)]+1,∞}]

FullSimplify[%,0<p<1]

%/.p->1/2

Numerical Check

MeanDeviation[RandomInteger[GeometricDistribution[.2],10^6]//N]

2 (((1-p))^(Floor[(1/p)])) Floor[(1/p)]/.p->.2

MeanDeviation[RandomInteger[GeometricDistribution[.5],10^6]//N]

2 (((1-p))^(Floor[(1/p)])) Floor[(1/p)]/.p->1/2

HypergeometricDistribution

pdf=Simplify[PDF[dist=HypergeometricDistribution[n,m,N],k],α>1]

m=Simplify[Mean[dist],α>1]

Sum[-pdf(k-m),{k,0,Floor[m]}]+Sum[pdf(k-m),{k,Floor[m]+1,∞}]

LogSeriesDistribution

?LogSeriesDistribution

pdf=PDF[dist=LogSeriesDistribution[θ],n]

m=Mean[dist]

s=Sum[pdf Abs[n-m],{n,1,∞}]

FullSimplify[Sum[((θ^n)/(n Log[1-θ])) (n+(θ/((1-θ) Log[1-θ]))),{n,Floor[(θ/((1-θ) Log[1-θ]))]}]+Sum[-((θ^n)/(n Log[1-θ]))(n+(θ/((1-θ) Log[1-θ]))),{n,Floor[(θ/((1-θ) Log[1-θ]))]+1,∞}],0<θ<1]

NegativeBinomialDistribution: (p^(N-1)) (((1-p))^((Floor[N ((1/p)-1)])+1)) ((Binomial[μ, N-1]) ((-N+p μ+p) (Hypergeometric2F1[1, μ+1, -N+μ+2, 1-p])-N+μ+1)-(p-1) p (Binomial[μ+1, N-1]) (Hypergeometric2F1[μ+2, 2, -N+μ+3, 1-p]))

pdf=PDF[dist=NegativeBinomialDistribution[N,p],n]

m=Mean[dist]

FullSimplify[Sum[-(((1-p))^n) (p^N) Binomial[-1+n+N,-1+N](n-(((1-p)N)/p)),{n,0,Floor[(((1-p)N)/p)]}]+Sum[(((1-p))^n) (p^N) Binomial[-1+n+N,-1+N](n-(((1-p)N)/p)),{n,Floor[(((1-p)N)/p)]+1,∞}],0<p<1&&N>0&&N∈Integers]

%/.Floor[N/p]:>μ

TraditionalForm[%]

Numerical Check

MeanDeviation[RandomInteger[NegativeBinomialDistribution[10,.2],10^7]//N]

(((1-p))^(1+Floor[(-1+(1/p))N])) (p^(-1+N)) (Binomial[Floor[(N/p)],-1+N] (1-N+Floor[(N/p)]+(-N+p+p Floor[(N/p)]) Hypergeometric2F1[1,1+Floor[(N/p)],2-N+Floor[(N/p)],1-p])-(-1+p) p Binomial[1+Floor[(N/p)],-1+N] Hypergeometric2F1[2+Floor[(N/p)],2,3-N+Floor[(N/p)],1-p])/.{p->.2,N->10}

PoissonDistribution:((2 (e^(-μ)) (μ^((Floor[μ])+1)))/((Floor[μ])!))

?PoissonDistribution

pdf=PDF[PoissonDistribution[μ],n]

m=Mean[PoissonDistribution[μ]]

s=Sum[pdf Abs[n-m],{n,0,∞}]

FullSimplify[Sum[-((Exp[-μ])/(n!))(μ^n)(n-μ),{n,0,Floor[μ]}]+Sum[((Exp[-μ])/(n!))(μ^n)(n-μ),{n,Floor[μ]+1,∞}],μ>0]

Numerical Check

MeanDeviation[RandomInteger[PoissonDistribution[.2],10^6]//N]

((2 (E^(-μ)) (μ^(1+Floor[μ])))/(Floor[μ]!))/.μ->.2

ZipfDistribution:((2 ((Zeta[ρ+1]) (HurwitzZeta[ρ, (Floor[((Zeta[ρ])/(Zeta[ρ+1]))])+1])-(Zeta[ρ]) (HurwitzZeta[ρ+1, (Floor[((Zeta[ρ])/(Zeta[ρ+1]))])+1])))/((Zeta[ρ+1])^2))

?ZipfDistribution

pdf=PDF[dist=ZipfDistribution[ρ],n]

m=Mean[dist]

s=Sum[pdf Abs[n-((Zeta[ρ])/(Zeta[1+ρ]))],{n,1,∞}]

Sum[-((n^(-1-ρ))/(Zeta[1+ρ])) (n-((Zeta[ρ])/(Zeta[1+ρ]))),{n,1,Floor[((Zeta[ρ])/(Zeta[1+ρ]))]}]+Sum[((n^(-1-ρ))/(Zeta[1+ρ])) (n-((Zeta[ρ])/(Zeta[1+ρ]))),{n,Floor[((Zeta[ρ])/(Zeta[1+ρ]))]+1,∞}]

FullSimplify[%,ρ>0]

Numerical Check

MeanDeviation[RandomInteger[ZipfDistribution[2.2`10],10^5]//N]

(1/((Zeta[1+ρ])^2))2 (-HurwitzZeta[1+ρ,1+Floor[((Zeta[ρ])/(Zeta[1+ρ]))]] Zeta[ρ]+HurwitzZeta[ρ,1+Floor[((Zeta[ρ])/(Zeta[1+ρ]))]] Zeta[1+ρ])/.ρ->2.2`20