Normal III

 

A Wonderful Guy

I'm as corny as Kansas in August,
I'm as normal as blueberry pie.
No more a smart little girl with no heart,
I have found me a wonderful guy!

One Blueberry Pie distribution coming right up!

In a normal distribution, the mean is equal to the median. (Or as Donald Trump Junior so famously put it “That’s how math works”).  However the mean is not always equal to the median in many distributions.  This is often only because the negative numbers are not reported because of where the y-axis is being set.  For example saying that temperatures are 40 below zero is only because the temperature is being measured on a relative scale, not on an absolute scale.  If an absolute scale had been used, then the median should be equal to the mean for a normal distribution.  Stating the median and the mean with a with a relative rather than an absolute zero, allows the calculation of the variance, if negative values had been allowed

NAR, the National Association of Realtors, has reported the mean ($639,000) and median ($396,500) prices for existing home sales.  Since the mean is not equal to the median this is hardly normal. It is suggested that this might be because negative sales prices are obviously not reported.  But if they could be reported, then if this was a normal distribution then no values would be observed for the mean minus 3 times the square root of the variance.  This requires that the variance must be the mean²/9=45.37 million.  It has already been reported that 50% of the homes sell for less than $396,500. If this were also a normal distribution, then 68% of the existing homes should sell for less than $852,000;  95% of the existing homes should sell for less than $1,065,000 and 100% of the existing homes should sell for less than  $1,278,000.  If there are existing homes selling for more than $1.3 million, then that also indicates that the existing home sales prices are not normal.

Pro-Choice II

 

My Way

And now the end is here
And so I face that final curtain
My friend I'll make it clear
I'll state my case, of which I'm certain
I've lived a life that's full
I traveled each and every highway
And more, much more
I did it, I did it my way

Yes, but did you also let me do it my way, or did you insist that I do it your way?

Those who are Pro-Life think they are going to heaven.  If they are Pro-Life but also Anti-Choice, then I am afraid that they have violated the First of the Ten Judeo-Christian Commandments, “I am the Lord thy God . Thou shalt have no other gods before me.”  By being Anti-Choice, they have confused a discontinuity with the absolute.  They are saying that Good is equal to Evil, that they have chosen Good, and everyone else should have no choice/should also choose Good.  Unfortunately the math does not work that way.

An exponential distribution has a discontinuity.  Traditionally this is zero, but it can be shifted to any discontinuity, µ.  While the exponential distribution is not defined before that discontinuity, it can be reflected at that discontinuity. This is implicitly saying that if the distribution at the discountinuity has a value of 144%, ln(2)%, certainty, then there is a pseudo mean, because of the discontinuity, at µ+1/λ.  That is interesting but if you smooth out the discontinuity to a normal distribution, and say that at most you can be 100% certain, then a factoring of a normal distribution is appropriate. But by saying it must be 100% certain, you are also saying that there is a Cumulative Distribution Function, CDF, of that factoring, which has almost the shape as the CDF of the exponential distribution with a discontinuity.  But the problem is that before the discontinuity, the exponential distribution implies a negative absolute while the factored continuous normal distribution has only one absolute. 

If you say that there is a choice, then you should only have a 50% chance of making that choice at the discontinuity. However by also saying that choices made before the discontinuity should not be considered, are treated as if they are zero, the Cumulative Distribution Function is of course substantially lower, and it takes until well after the discontinuity to reach 100%.  If you are Pro-Life AND Pro-Choice, then your CDF will reach 100% much sooner. If you are Anti-Choice, then you are also saying that there are two absolutes, not one.

So if someone is Pro-Life AND Pro-Choice, then they are keeping the Commandments, not breaking them.

The Unethical Justice Alito

 

Waiting for the Robert E. Lee.

Way down on the levy in old Alabamy
There's Daddy and Mammy
There's Ephraim and Sammy
On a moonlight night you can find them all
While they are waiting,
The banjos are syncopating

Stop the music “Strip Search” Sammy! Your waiting is over!

You just knew that Justice “Strip Search Sammy” Alito could not let Justice “Long Dong Silver” Thomas get all of the headlines as the villain of the Supreme Court. “Strip Search Sammy” just wrote an editorial in the Wall Street Journal that it was unconstitutional for Congress to impose a code of ethics on the Supreme Court. Let’s give you a civics lesson “Strip Search.”  The Constitution, as in says in the very opening  line, is the document of “We The People.”  The Congress is the elected representatives of “We The People", whose job is to enact laws. The Supreme Court only offers opinions on those laws as to whether Congress acted against the stated protections, not of the Supreme Court, but of “We The People.’  If Congress says the Supreme Court should be 

1. ethical, or 
2. have terms limits because its justices are not immortal, or 
3. should decide cases not by domination but by certainty, or 
4. decide that there should be ten rather than the current nine members of the Supreme Court, as has been the case in the past, 

AND that law does not require an action that is specifically prohibited by the Constitution,  (And let’s save you the trouble, “Strip Search”.  Just as the Constitution did not mention a right to an abortion,  it also does NOT prohibit ethical standards), then who are you to speak.  Put down the shovel, Sammy. You are in a deep enough hole right now. Your waiting is over. The People are coming for you.

Pythagoras

 

Supercalifragilisticexpialidocious

He traveled all around the world and everywhere he went
He'd use his word and all would say there goes a clever gent
When dukes or Maharajas pass the time of day with me
I say me special word and then they ask me out to tea (woo)

But if you say HYPERBOLICfragilisticexpialidocious, then you might be even smarter!

e^ix=cos(x)+isin(x), where cosh(x) is the hyperbolic version of the trigonometric cosine, cos(x), is Euler’s Formula, where i is  the imaginary number, √-1, which is called j by electrical engineers.  This can be restated as y=f(x)=ln(cos(x)+i*sin(x))/i, and which implies cos(x)=e^ix - isin(x). If f(x) is coordinate transformed by rotating by 90° it becomes x=f(y)=ln(cos(y)+i*sin(y))/i. This can restated as y=g(x) where g(x)=sin(90°)f(x).  But this also means that cosh(ix)= cos(y) + isin(y). Any cosine can also restated as √(1-sin²). This means that cosh(ix)=cos(x)+isin(x) therefore becomes cosh(ix)=√(1 - sin²(x))+isin(x) and also cosh(ix)=√(1-sin²(y)+2isin(y)). If only the real components of the complex numbers are used, this means that the rotation of Euler’s Formula by 90° is true only if y and x are identical and zero.

Pythagoras’ Theorem, c=√(a²+b²), is true because cos²(x)+sin²(x)=1, which is also the formula for a circle.  By contrast, cosh²(x)-sinh²(x)=1 is the formula for a hyperbola.  Pythagoras’ Theorem is only true for a flat surface.  If the surface is spherical, whose sphere has a radius R, then the formula for the hypotenuse, c, of a right triangle whose other sides are a and b is cos(c/R)=cos(a/R)cos(b/R).  If this is expressed as a series, then as R approaches infinity, the series becomes c=√(a²+b²). For a hyperbolic surface, the formula for a hypotenuse is cosh(c)=cosh(a)cosh(b), but if the coefficient of the imaginary number is zero, or only the real portion is used, then this also becomes Pythagoras’ Theorem. 

For a finite spherical surface, as R, the radius of the sphere, becomes very large compared to a or b, this becomes Pythagoras’ Theorem.  On a infinte surface, it also becomes Pythagoras’ Theorem but we appaer t only use the ral portionof the solution. Using only the real portion of a complex solution is precisely how electrical engineers treat the imaginary portion when their solution is a complex number. And now you know the rest of the story. If you use only Pythagoras' Theorem, then apparently you have no imagination!  If you have an imagination then you are HYPERBOLICfragilisticexpialidocious!

Distributions

 

Ain’t We Got Fun

There's nothing surer
The rich get rich and the poor get poorer
In the meantime, in between time
Don't we have fun?

But is it normal for the rich to get richer?

It is proposed that the Cumulative Distribution Function, CDF, for an exponential distribution, which is 1-e^-λx, with a rate parameter, λ, can be approximated by a coordinate translation of the random normal logistics distribution, also known as the hyperbolic secant squared distribution, whose CDF is ½*tanh((x-µ)/(2*s))+½, from an origin of (0,0) to an origin of (λ, 0.5) if that random normal CDF is also scaled by 2. This means that the range parameter, s, of the logistics distribution can be approximated by 1/(2*λ*ln(2)). While the exponential distributions is traditionally only defined for x>0, this can be translated to begin at any location, µ, if the exponential distribution is also defined for x>µ>0.

Because the logistics function is already defined for all ranges of x, this means that the exponential distribution, whose CDF is also known as the exponential association, can also be defined for all values of  x, including x<µ, if its parameter s is a function of λ. This means that there is no need for a combination of the exponential distribution and a random normal function, either as an Exponentially Modified Gaussian distribution as proposed by Grushka [1], or as an Exponentially Modified Logistic distribution as proposed by Reyes [2]

The figure below shows the CDF of a logistics distribution (blue), which does not look like the CDF of the exponential distribution (red). Also shown as a dash red curve is what the CDF for the exponential distribution would be for x<0. The doubling of logistics function with a shift along the y-axis of the origin from (0, 0) to an origin of (0, 0.5) does look like the exponential distribution for x>0 (green).

As shown below, if the curves are shifted on the x-axis to both cross at µ, by shifting the exponential distribution from an origin of (0, 0) to an origin of (µ, 0) then the two curves look more similar for x>µ.

By setting the two curves equal at a common location, µ, it is possible to solve for s, the range parameter of the logistics distribution in terms of λ, the rate parameter of the exponential distribution. This function is s=1/(2*ln(2)*λ). If the variance is equal to 1.0, then the relationship between s and the variance, σ², as s²π²/3 can be used to compute that s=0.55. At that value of s, this means that the correlation between the two curves from µ to µ+3σ, is almost perfect at 0.9967. However if the difference between that scaled logistics distribution greater than the median and the exponential distribution is set to a minimum, the values become λ= 1/ln(2)=1.44, s =0.5, the variance thus becomes 0.822 and the correlation between the exponential and the logistics curve, scaled and shifted, increases to 0.9982.

It is thus proposed that there is no need to develop a new distribution combining the exponential and a random normal distribution. The exponential distribution with a constraint of x>µ, appears to be merely the upper half of a normal logistics distribution, the half beginning at the median. It is also suggested the lowest variance for a normal distribution should be 0.822, the lowest standard devaiation should be 0.9069, s should be 0.5, and that the rate parameter of the exponential distribution is related to the difference between the mean and median of any distribution.

Thus if the mean household income in 2021 is $66,018 and the median household income is $58,153 according to the U.S. Census, and income follows an exponential distribution, the curve would be as shown below, which also shows the reported mean household income by the mid‑point of a decile, as well as the reported mean income limit of the highest 5%. This suggests that only when zero represents an absolute value, e.g. as the vector distance from an object, or an empty condition, where the mean and the median of the distribution are the same, will this be a true exponential distribution. It will be skewed by definition and is not normal. However if the median and the mean are appreciably different, then the distribution may only appear to follow an exponential distribution, but the distribution is in fact normal and its appearance as a skewed exponential distribution is because only the portion above the median is being used. Or as Garrison Keillor ironically puts it in his tales from Lake Wobegon, “All the children are above average.”

The chart above has been adjusted for inflation, i.e. all incomes are in 2021 US Dollars.  Both the 1968 and the 2021 distributions have the same total income for society but only vary in how it is distributed to individual households.  It suggests that, the income distribution in 1968 was less skewed, and that if it was viewed as a normal distribution for all incomes, including subsidies and transfers, i.e. negative incomes, the lower income range would be between $0 to $100,645 instead of the current range of $0 to $163,547 and the income to be wealthy would be $301,934 instead of $490,642.  The 1968 distribution was less normal, had a lower coefficent of determination, r², to the random distribution, but was more equitable, had a lower variance. The 2021 distribution was more normal but less equitable. The challenge is to distribute incomes in a manner that is both normal and equitable.

[1] Grushka, E. (1972). Characteristics of Exponentially Modified Gaussian Peaks in Chromatography. Analytical Chemistry Vol 44, pp. 1733-1738.

[2] Reyes, J., Venegas, O., & Gómez, H. W. (2018). Exponentially-Modified Logistic Distribution with Application to Mining and Nutrition Data. Applied Mathematics & Information Sciences Vol 12 Number 6, pp. 1109-1116.

 

 

Monism vs Dualism

 

All or Nothing At All

All or nothing at all
Nothing at all
There ain't nothing at all
Nothing at all

Which is NOT all, or the opposite of all.

Yesterday was “All or Nothing” Day, which I missed.  However it does allow me the opportunity to point out that All or Nothing at all, ∞ or 0, is different than All or Its opposite, ∞ or -∞.  The first is monism and there is only one absolute.  The second is dualism and there are two absolutes, one positive and one negative.

The integral of All or Nothing at All is different than the integral of All or The Opposite of All.  The first is also the Cumulative Distribution Function of a normal random distribution, which takes on values between zero and 1, and from -∞ to ∞ has an integral of 1.  The second is the integral of a hyperbolic tangent function which take on values between -1 and 1, and has an integral between -∞ and ∞ of 0. Saying that there are two absolutes is thus fundamentally different than saying that there is only one absolute.

Choice VI

 

What'd I Say, Pt. 1

See the girl with the red dress on
She can do the Birdland all night long, yeah, yeah
What'd I say?
All right
Well, tell me what'd I say

I want to choose what I want, not what you say!

The moral imperative is strong in the United States.  Whether it is religion, alcohol, pornography, sex work, drug use, birth control, abortion, miscegenation, misogyny, etc., those who have chosen to not use or do something feel superior to those who might choose to use or do it.

But if there is one choice, Good and no choice, Not Good, which is different than Good and Evil which is two choices, Good, Evil and no choice, Not Good/Not Evil, then forcing your choice on others results in fewer individuals in the same time making the same choice that you did.

Regardless of whether you force everyone to make your choice, or you let them make their own choice, ultimately everyone will make the same choice.  It is just that mathematically you can show that more individuals will make that choice if they are not forced.  Forcing a choice gets to that choice at a slower rate.  Changing the variance, the range of choices, does not change the Cumulative Distribution Function where everyone is forced to make one choice, an exponential distribution.  However increasing, or decreasing, the variance, σ², whose square root goes by the term standard deviation, only changes the shape of the cumulative random choice slightly.  It appears that thinking that this is the only choice confuses the choice with the variance.  However if only one choice is available, then the range, s, which is half of that choice, and the variance is 0.822 not 1.  In fact it is not until s=1, which implies two choices not one, that the random continuous curve approaches the exponential distribution curve which has a discontinuity at a location, µ.

The point I am trying to make is that it is the destination, not the path to reach that destination, that is important.  Acting like your path is the only path ignores those who reached that destination before you, but chose a different path.  And it ignores those who would have reached that destination after you but might have chosen a different path.  Acting like there is only one destination, when in fact this is more similar to the results of choosing that destination AND its opposite, a hyperbolic tangent, rather that one destination and no destination at all.

So by forcing everyone to choose your path, you are making it slower for everyone to reach the same destination as you; and you are implicitly acknowledging that there are two destinations, not one.  You are making not an argument between pro-choice and no choice.  You are making the argument between two opposing choices.  You are actually arguing for more standard deviation rather than less.

But will the math convince anyone?  LOL, the same people advocating for only their path are the same ones who tried to get the value of pi, π, changed to 3, because they thought otherwise the math was too hard.  Arguing for “no choice” is like arguing that 2+2=5. Why argue in favor of truth when there are alternatives facts, …uh lies….,  available?