It's Relative III

 Dead Man's Curve

(Dead Man's Curve) is no place to play

(Dead Man's Curve) you'd best keep away

(Dead Man's Curve) I can hear 'em say

Won't come back from Dead Man's Curve

If the universe is curved, what does that do to randoneess and choices?

Variance is the dispersion of the data, X,  in a statistical distribution. Mathematically its formula is

Var(X)=∑ (p_i*(x_i-µ))² for i =1 to n variables

The variance is traditionally defined as the square of a parameter of a random distribution,  σ², where σ is also known as the standard deviation, SD.  The other parameter of a random distribution, µ, the mean, which is equal to the median of a normal distribution, is

µ=∑ p_i(x_i-µ)*x_i for i=1 to n variables.

However it is not correct to say that the variance, which is the square of the standard deviation, SD, can be solved by Pythagoras’ formula. This is only true on a flat surface.  In fact on a flat surface, rather than a summation, this should be expressed as a product as

SD(X)=cos^-1(∏ cos(p_i(x_i-µ))) for i=1 to n variables.

On a flat surface, this is identical to  

SD(X)=√( (p_i(x_i-µ))² for i =1 to n variables or √( (x_i-µ))²/n).

That Bessel’s correction, n/(n-1), is necessary for the SD is an indication that the surface is not flat.

On a spherical surface, the formula should be

SD(X)=R*cos^-1(∏ cos(p_i(x_i-µ)/R)) for i=1 to n variables.

Here R is the radius of the spherical surface.  As R approaches infinity, e.g. the sphere become very large compared to any p_i(x_i-µ), this becomes identical to the formula for a flat surface.

On a hyperbolic surface, the formula should be

SD(X)=cosh^-1(∏ cosh(p_i(x_i-µ))) for i=1 to n

The variance, σ², of a random logistics distribution, where the parameters are µ and s, is s²*π²/3.  Because this is the summation of a square, it is correct to say that σ²=s²*π²/3, but it is NOT correct to say that because of this s=σ√3/π.  This is only true for a flat surface.

If the surface is hyperbolic, then the correct formula is

s=cosh^-1(cosh(σ√3/π))

Because cosh^-1(x)=ln(x±√(x²-1)), cosh²(x)-sinh²(x)=1, cosh(x)=½*(e^x+e^-x), and sinh(x)= ½*(e^x-e^-x), this can be expressed as

s=ln(cosh(σ√3/π) ±√( cosh²(σ√3/π)-1))

s=ln(cosh(σ√3/π) ± sinh(σ√3/π))

s=ln(½*(e^σ√3/π +e^-σ√3/π) ± ½*(e^σ√3/π -e^-σ√3/π))

s= σ√3/π ± σ√3/π= 2* σ√3/π or 0

This is because the variance, the square of the standard deviation, is a summation, NOT a single variable.

Only if the surface is flat is it true that s=σ√3/π.

This means that the random normal logistics distribution, the hyperbolic secant distribution,  

PDF = (1/(4s))*sech²((x-µ)/2s); CDF = ½*tanh((x-µ)/2s)+ ½

can be expressed in terms of the parameters µ and σ as

PDF = (π /(8* σ√3))*sech²(π *(x-µ)/(2* σ√3)); CDF = ½*tanh( π *(x-µ)/(2* σ√3))+ ½.

When the standard deviation, σ, is π/(4√3) then this simplifies to

PDF = ½*sech²(x-µ); CDF = ½*tanh(x-µ)+ ½ which is identical to saying that  s=½.

In this case, when x=µ, then the PDF is 50% and the CDF is also 50%. If π is the absolute, and negative numbers are not allowed, then, according to Pearson’s Second Coefficient of Skewness, µ=π/2.  If s=½, then σ²=(½)²π²/3=0.822467.

If the universe is hyperbolic, as proposed by Mabkhout (Mabkhout, 2012), the absolute is defined as π, x is an absolute measurement which means that 0 is an absolute zero, numbers x<0 are not defined, AND the universe is random, then if the universe is normal, it has both a mean and a variance which can be expressed in terms of its absolute. That universe also allows choice, s=½, where there is one choice: the absolute or absolute zero.

If the universe is hyperbolic and random, then random events do not repeat except on an imaginary plane. Random events may appear to repeat, be cyclical, but it is not in the real surface where the coefficient of the imaginary axis is 0.  I.e., Mark Twain was correct.  History does NOT repeat, but it sure does rhyme.

Mabkhout, S. (2012). The infinite distance horizon and the hyperbolic inflation in the hyperbolic universe. Phys. Essays, 25(1), p.112.