Descriptive Statistics

Calculate and verify statistical descriptors

Max: `hat(x)`
Min: `ul (x)`
Average: `bar(x)`
Variance: `bar(bar(x))`

for parameters of routes

Steps (Total stopping time): `n`
Height: `u`
Odd steps: `o`
Even Steps: `e`
Completeness: `c`
Finesse: `gamma`

at given

Total Stopping Time : `n`
Height : `u`

Results are in the tables bellow, demonstrations follow

At `n`	Minimum	Maximum	Average	Variance
`o`	`0`	`hat(o)_n`	`n/4`	`(3*n)/16`
`e`	`n-hat(o)_n`	`n`	`(3*n)/4`	`(3*n)/16`
`u`	`n*ln(2)-hat(o)_n*ln(3)`	`n*ln(2)`	`K_1*n`	`(3*n*ln(3)^2)/16`
`c`	`0`	`hat(o)_n/n`	`1/4`	`3/(16*n)`
`gamma`	`1/ln(2)`	`n/(n*ln(2)-hat(o)_n*ln(3))`	`~~(4*K_1^2*n+12*ln(3)^2)/(K_1^3*n^2)`	`~~(48*ln(3)^2)/(K_1^4*n)`

`K_1=(4*ln(2)-ln(3))/4`

At `u`	Minimum	Maximum	Average	Variance
`n`	`u/ln(2)`	`(u+hat(o)_u*ln(3))/ln(2)`	`C_1*u`	`C_1^3*ln(3)/4*u`
`o`	`0`	`hat(o)_u`	`C_1*u/2`	`C_1^3*ln(2)^2/(4*ln(3))*u`
`e`	`u/ln(2)`	`(u+hat(o)_u*(ln(3)-ln(2)))/ln(2)`	`C_1*u/2`	`C_1^3*(ln(3)-ln(2))^2/(4*ln(3))*u`
`c`	`0`	`(hat(o)_u*ln(2))/(u+hat(o)_u*ln(3))`	`~~1/2-1/(4*u)`	`~~1/(4*C_1*u*ln(3))`
`gamma`	`1/ln(2)`	`(u+hat(o)_u*ln(3))/(ln(2)*u)`	`C_1`	`C_1^3*ln(3)/(4*u)`

`C_1=2/(2*ln(2)-ln(3))`

At given Total Stopping Time

Experimental (real) data are created by contructing Total Stopping Time Classes `R_k`up to class `R_(50)` by
`R_0={1}, R_1={2}, R_3={4}`
`R_(k+1)={T(x_i in R_k)}uu{B(x_i in R_knn[dot2]_3)}`

Number of values

the model is
`r_(n,o)=(((o),(n)))/(3^o)`
`r_n=sum_(o=0)^nr_(n,o)=(4/3)^n`

Odd Steps

Minimum

`AAn, EEz=2^n, sigma_(oo)(z)=n^^o(z)=0`
`rArr ul(o)_n=0`
 

Maximum

As already shown in the Total Stopping Time Chapter
`hat(o)_n` is solution of `((hat(o)_n),(n))=3^(hat(o)_n)` 

Average

`S_(0,o)=r_n=sum_(o=0)^nr_(n,o)=sum_(o=0)^n((((o),(n)))/(3^o))=(4/3)^n`
`S_(1,o)=sum_(o=0)^n(r_(n,o)*o)=sum_(o=0)^n((((o),(n)))/(3^o)*o)=(4^(n-1)*n)/(3^n)`
`bar(o)_n=S_(1,o)/S_(0,o)=n/4`
 

Variance

`S_(2,o)=sum_(o=0)^n(r_(n,o)*o^2)=sum_(o=0)^n((((o),(n)))/(3^o)*o^2)=(4^(n-2)*(n^2+3*n))/(3^n)`
`bar(bar(o))_n=(S_(0,o)*S_(2,o)-S_(1,o)*S_(1,o))/(S_(0,o))^2=(3*n)/16`

Even Steps

with :`n=o + e`

Minimum

`ul(e)_n=n-hat(o)_n`
 

Maximum

`hat(e)_n=n-ul(o)_n=n`
 

Average

`bar(e)_n=n-bar(o)_n=n-n/4=(3*n)/4`
 

Variance

`S_(0,e)=S_(0,o)=(4/3)^n`
`S_(1,n)=sum_(o=0)^n(r_(n,o)*n)=n*r_n=n*(4/3)^n`
`S_(2,n)=sum_(o=0)^n(r_(n,o)*n^2)=n^2*r_n=n^2*(4/3)^n`
`S_(1,e)=sum_(o=0)^n(r_(n,o)*e)=sum_(o=0)^n(r_(n,o)*(n-o))=S_(1,n)-S_(1,o)`
`S_(1,e)=n*(4/3)^n-(4^(n-1)*n)/(3^n)=n*(4/3)^(n-1)`
`S_(2,e)=sum_(o=0)^n(r_(n,o)*e^2)=sum_(o=0)^n(r_(n,o)*(n-o)^2)=S_(2,n)+S_(2,o)-2*n*S_(1,o)`
`S_(2,e)=n^2*(4/3)^n+(4^(n-2)*(n^2+3*n))/(3^n)-2*n*(4^(n-1)*n)/(3^n)=(4/3)^n*((3*n+9*n^2)/16)`
`bar(bar(e))_n=(S_(0,e)*S_(2,e)-S_(1,e)*S_(1,e))/(S_(0,e))^2=(3*n)/16`

Height

with :`u=n*ln(2)-o*ln(3)`

Minimum

`ul(u)_n=n*ln(2)-hat(o)_n*ln(3)`
 

Maximum

`hat(u)_n=n*ln(2)-ul(o)_n*ln(3)=n*ln(2)`
 

Average

`bar(u)_n=n*ln(2)-bar(o)_n*ln(3)=n*(4*ln(2)-ln(3))/4`
 

Variance

`S_(0,u)=S_(0,o)=(4/3)^n`
`S_(1,u)=sum_(o=0)^n(r_(n,o)*(n*ln(2)-o*ln(3)))=S_(1,n)*ln(2)-S_(1,o)*ln(3)`
`S_(1,u)=(4/3)^n*(n*(4*ln(2)-ln(3)))/4`
`S_(2,u)=sum_(o=0)^n(r_(n,o)*(n*ln(2)-o*ln(3))^2)=S_(2,n)*ln(2)^2+S_(2,o)*ln(3)^2-2*ln(2)*ln(3)n*S_(1,o)`
`S_(2,u)=ln(2)^2*(4/3)^n*n^2+ln(3)^2*(4/3)^n*((3*n+n^2)/16)-2*ln(2)*ln(3)*n*(n/4)*(4/3)^n`
`S_(2,u)=(4/3)^n*((3*n*ln(3)^2+n^2*(16*ln(2)^2+ln(3)^2-8*ln(2)*ln(3)))/16)`
`S_(2,u)=(4/3)^n*((3*n*ln(3)^2+n^2*(4*ln(2)-ln(3))^2)/16)`
`bar(bar(u))_n= (S_(0,u)*S_(2,u)-S_(1,u)*S_(1,u))/(S_(0,u))^2 =(3*n*ln(3)^2)/16`

Completeness

with :`c=o/n`

Minimum

`ul(c)_n=ul(o)_n/n=0`
 

Maximum

`hat(c)_n=hat(o)_n/n`
 

Average

`bar(c)_n=bar(o)_n/n=1/4`
 

Variance

`bar(bar(c))_n=bar(bar(o))_n/(n^2)=3/(16*n)`

Finesse

with :`gamma=n/u=1/(ln(2)-c*ln(3))`

Minimum

`ul(gamma)_n=1/(ln(2)-ul(c)_n*ln(3))=1/ln(2)`
 

Maximum

`hat(gamma)_n=n/(n*ln(2)-hat(o)_n*ln(3))`
 

Average

`gamma=n/u=n*(1/u)`
with `bar((1/u))~~1/bar(u)+bar(bar(u))/bar(u)^3`
`bar(u)=K_1.n` with `K_1=(4*ln(2)-ln(3))/4`
`bar(gamma)_n~~(4*K_1^2*n+12*ln(3)^2)/(K_1^3*n^2)`
 

Variance

`gamma=n/u=n*(1/u)`
with `bar(bar((1/u)))~~bar(bar(u))/bar(u)^4`
`bar(bar(gamma))_n~~(48*ln(3)^2)/(K_1^4*n)`

At given Height

As we cannot get significant estimates parameters around `u`, we use cumulated quantities from `1` to `e^u` instead, noted as capitals, and use the following corrections
`overline(X)(xi)-xi*(del overline(X)(xi))/(del xi)=overline(x)(xi)`
`bar(bar(X))(xi)+xi*(del bar(bar(X))(xi))/(del xi)-xi^2*((del bar(X)(xi))/(del xi))^2=bar(bar(x))(xi)`
Quantities have been evaluated up to `2^(28)`

Summation values

using `u=n*ln(2)-o*ln(3)` to link `n`, `o` and `e` to u, we cannot use discrete summations, we use continous representation instead with results dependant of the parameter used.

with `n`:
`r_(u,n)=((((n*ln(2)-u)/ln(3)),(n)))/(3^((n*ln(2)-u)/ln(3)))`
`S_(0n)=int_(o=0)^(oo)r_(u,n)*dn=A_(0n)*e^u`
`S_(1n)=int_(o=0)^(oo)r_(u,n)*n*dn=(A_(1n)*u+B_(1n))*e^u`
`S_(2n)=int_(o=0)^(oo)r_(u,n)*n^2*dn=(A_(2n)*u^2+B_(2n)*u+C_(2n))*e^u`

with `o`:
`r_(u,o)=(((o),((u+o*ln(3))/ln(2))))/(3^((u+o*ln(3))/ln(2)))`
`S_(0o)=int_(o=0)^(oo)r_(u,o)*do=A_(0o)*e^u`
`S_(1o)=int_(o=0)^(oo)r_(u,o)*o*do=(A_(1o)*u+B_(1o))*e^u`
`S_(2o)=int_(o=0)^(oo)r_(u,o)*o^2*do=(A_(2o)*u^2+B_(2o)*u+C_(2o))*e^u`

with `e`:
`r_(u,e)=((((e*ln(2)-u)/(ln(3)-ln(2))),((e*ln(3)-u)/(ln(3)-ln(2)))))/(3^((e*ln(2)-u)/(ln(3)-ln(2)))`
`S_(0e)=int_(o=0)^(oo)r_(u,e)*de=A_(0o)*e^u`
`S_(1e)=int_(o=0)^(oo)r_(u,e)*e*de=(A_(1e)*u+B_(1e))*e^u`
`S_(2e)=int_(o=0)^(oo)r_(u,e)*e^2*de=(A_(2e)*u^2+B_(2e)*u+C_(2e))*e^u`

Total Stopping Time

`A_(0n)=C_1*ln(3)`
`A_(1n)=C_1^2*ln(3)`
`B_(1n)=C_1^3*ln(3)^3/4`
`A_(2n)=C_1^3*ln(3)`
`B_(2n)=C_1^4*(3*ln(3)^3)/4`
`C_(2n)=C_(2n)`

with `n=(u+o*ln(3))/ln(2)`

Minimum

`ul(n)_u=u/ln(2)`

Maximum

`hat(n)_u=(u+hat(o)_u*ln(3))/ln(2)`

Average

`bar(n)_u=S_(1n)/S_(0n)`
`bar(n)_u=(A_(1n)*u+B_(1n))/A_(0n)`
`bar(n)_u=A_(1n)/A_(0n)*u+B_(1n)/A_(0n)`
`bar(n)_u=C_1*u+o(u) rarr C_1*u `
 

Variance

`bar(bar(n))_u=(S_(2,n)*S_(0n)-S_(1n)^2)/S_(0n)^2`
`bar(bar(n))_u=(A_(2,n)*A_(0n)-A_(1n)^2)/A_(0n)^2*u^2+(B_(2n)*A_(0n)-2*A_(1n)*B_(1n))/A_(0n)^2*u+(C_(2n)*A_(0n)-B_(1n)^2)/A_(0n)^2`
with:
`A_(2n)*A_(0n)-A_(1n)^2=C_1*ln(3)*C_1^3*ln(3)/4-(C_1^2*ln(3)/2)^2=0`
`bar(bar(n))_u=(B_(2n)*A_(0n)-2*A_(1n)*B_(1n))/A_(0n)^2*u+o(u)`
`bar(bar(n))_u rarr C_1^3*ln(3)^2/4*u`

Odd Steps

`A_(0o)=C_1*ln(2)`
`A_(1o)=C_1^2*ln(2)/2`
`B_(1o)=C_1^3*(ln(2)^2*ln(3))/4`
`A_(2o)=C_1^3*ln(2)/4`
`B_(2o)=C_1^4*(ln(2)^3+ln(2)^2*ln(3))/4`
`C_(2o)=C_(2o)`

Minimum

`ul(o)_u=0`

Maximum

`hat(o)_u` is solution of  : `{(u=n*ln(2)-o*ln(3)),(((n),(o))=3^o):}`

Average

`bar(o)_u=S_(1o)/S_(0o)`
`bar(o)_u=(A_(1o)*u+B_(1o))/A_(0o)`
`bar(o)_u=A_(1o)/A_(0o)*u+B_(1o)/A_(0o)`
`bar(o)_u=C_1/2*u+o(u) rarr C_1/2*u `

Variance

`bar(bar(o))_u=(S_(2,o)*S_(0o)-S_(1o)^2)/S_(0o)^2`
`bar(bar(o))_u=(A_(2,o)*A_(0o)-A_(1o)^2)/A_(0o)^2*u^2+(B_(2o)*A_(0o)-2*A_(1o)*B_(1o))/A_(0o)^2*u+(C_(2o)*A_(0o)-B_(1o)^2)/A_(0o)^2`
with:
`A_(2o)*A_(0o)-A_(1o)^2=C_1*ln(2)*C_1^3*ln(2)/4-(C_1^2*ln(2)/2)^2=0`
`bar(bar(o))_u=(B_(2o)*A_(0o)-2*A_(1o)*B_(1o))/A_(0o)^2*u+o(u)`
`bar(bar(o))_u rarr C_1^3*ln(2)^2/4*u`

Even Steps

`A_(0e)=C_1*(ln(3)-ln(2))`
`A_(1e)=C_1^2*(ln(3)-ln(2))/2`
`B_(1e)=C_1^3*((ln(3)-ln(2))^2*ln(3))/4`
`A_(2e)=C_1^3*(ln(3)-ln(2))/4`
`B_(2e)=C_1^4*(2*ln(3)^3-ln(2)^3+4*ln(2)^2*ln(3)-5*ln(2)*ln(3)^3)/4`
`C_(2e)=C_(2e)`

Minimum

`ul(e)_u=u/ln(2)`

Maximum

`hat(e)_u=(u+hat(o)*(ln(3)-ln(2)))/ln(2)`

Average

`bar(e)_u=S_(1e)/S_(0e)`
`bar(e)_u=(A_(1e)*u+B_(1e))/A_(0e)`
`bar(e)_u=A_(1e)/A_(0e)*u+B_(1e)/A_(0e)`
`bar(e)_u=C_1/2*u+o(u) rarr C_1/2*u `
 

Variance

`bar(bar(e))_u=(S_(2,e)*S_(0e)-S_(1e)^2)/S_(0e)^2`
`bar(bar(e))_u=(A_(2,e)*A_(0e)-A_(1e)^2)/A_(0e)^2*u^2+(B_(2e)*A_(0e)-2*A_(1e)*B_(1e))/A_(0e)^2*u+(C_(2e)*A_(0e)-B_(1e)^2)/A_(0e)^2`
with:
`A_(2e)*A_(0e)-A_(1e)^2=C_1*(ln(3)-ln(2))*C_1^3*(ln(3)-ln(2))/4-(C_1^2*(ln(3)-ln(2))/2)^2=0`
`bar(bar(e))_u=(B_(2e)*A_(0e)-2*A_(1e)*B_(1e))/A_(0e)^2*u+o(u)`
`bar(bar(e))_u rarr C_1^3*(ln(3)-ln(2))^2/4*u`

Completeness

with :`c=(ln(2)-1/gamma)/ln(3)`

Minimum

`ul(c)_u=0`

Maximum

`hat(c)_u=(hat(o)*ln(2))/(u+hat(o)*ln(3))`

Average

`bar(c)_u=1/ln(3)*(ln(2)-bar((1/gamma)))`
with
`bar((1/gamma))~~1/bar(gamma)+bar(bar(gamma))/bar(gamma)^3`
and
`{(bar(gamma) =C_1),(bar(bar(gamma))=(C_1^3*ln(3))/(4*u)):}`

`bar(c)_u~~1/2-1/(4*u)`

Variance

`bar(bar(c))_u=1/ln(3)^2*bar(bar((1/gamma)))`
with
`bar(bar((1/gamma)))~~bar(bar(gamma))/bar(gamma)^4`

`bar(bar(c))_u~~1/(4*C_1*u*ln(3))`

Finesse

with :`gamma=n/u`

Minimum

`ul(gamma)_u=ul(n)_u/u=1/ln(2)`

Maximum

`hat(gamma)_u=hat(n)_u/u=(u+hat(o)_u*ln(3))/(ln(2)*u)`

Average

`bar(gamma)_u=bar(n)_u/u=C_1`

Variance

`bar(bar(gamma))_u=bar(bar(n))_u/(u^2)=(C_1^3*ln(3))/(4*u)`