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?

Humor

 

I Started A Joke

I started a joke
Which started the whole world crying
But I didn't see
That the joke was on me, oh no

Tain’t funny, McGee

So what is funny? Humor is an incongruity, e.g. between a statement and an action, which is unintentional, or not intended as harm. A stuffy individual walking with dignity and slipping on a banana peel is funny. An insult that is not meant as an insult is funny, or at least that is what Don Rickles always said. An insult that was meant as an insult using the catch phrase from the Fibber McGee and Molly radio show, Tain’t funny. It is an insult.

Saying that an insult was not meant an insult but as a joke, does not make it less of an insult, It also does not mean that the insulted person can not take a joke. The ability to get a joke may be a way to discern whether someone is lying or telling the truth.

When Jonathan Swift wrote his book A Modest Proposal, he was intending humor, not writing a cookbook for the elimination of Irish children. Those who viewed it as a cookbook had to be reminded that it was humor. Unfortunately, this recently had to include Donald Trump’s lawyer in the E. Jean Carroll defamation lawsuit, where the judge felt compelled to remind Donald Trump’s lawyer that when E Jean Carroll called her book What Do We Need Men For? A Modest Proposal she was obviously writing humorously, not stating a personal opinion. When Marjorie Taylor Greene banged the gavel calling for order in the House chamber, the laughter that resulted was the incongruity of someone who flaunted House decorum, now calling for order. Saying that something is a joke does not make it a joke. Saying that a joke is a fact does not make it a fact. Being contrary to the facts is a lie, so reacting with laughter instead of anger, is being more than kind.

Changes

 

Crown of Creation

Life is change
How it differs from the rocks
I've seen their ways too often for my liking
New worlds to gain
My life is to survive
And be alive for you
No Man is an island. He’s a peninsula!

If you are pro-life, then you must also be pro-change!

Saying that life is change and that no man is an island means that mankind must be in favor of change and is a group animal not an individual animal. And as Shakespeare would say, “Therein lies the rub.”  Acting to oppose change, and acting as individuals, is easier. I personally do not like any change. I have been having the same breakfast at home almost every day for the last 20 years. I also like to pretend I am an individual and don’t like joining any groups. I don’t want to be a member of any group that would have me as a member. But acting to oppose change and like there are no groups of which you are not a part might be why we have problems today.

The current 435 members of the US House, and thus the current electoral college, might have been an acceptable compromise in 1911. Making it automatic might have been acceptable in 1941. However the world has changed in 2023 from what it was in either 1911, or 1941. Maybe the current method of reapportionment is no longer appropriate.

The world looks flat to an individual. But the world only appears locally flat but is actually round.

A current individual might not be able to construct the Great Pyramid. But that does not mean that it was built by aliens instead of a group of individuals.

Climate change happens over a very long time, probably longer than any individual’s lifetime, but that does not mean that it is not real for a group of individuals.  Just ask the ancient Mayans or the modern Pacific Islanders whether climate change is real.

I can’t make my own clothes, or even the cloth from which many clothes are made, but that does not mean that my ancestors were all unclothed. But that also does not mean that to survive as an individual, I must master all of those skills. Survivalists take note that the uncontacted “stone age tribes “ in the Amazon jungle may in fact be the descendants of survivalists from the Inca civilization.

Diseases can wipe out or change cultures in addition to killing individuals. Never mind COVID. Ask the indigenous Americans about smallpox or the Europeans about the Black Death.

Humans can influence climate change. The Mongol invasions in the 1200s probably brought about the Little Ice Age. https://umaine.edu/news/blog/2016/02/05/climate-change-and-the-rise-of-the-mongol-empire/

So opposing change and acting like only individuals matter is contributing to the problem, not a solution.

Bastille Day

 

La Marseillaise.

Aux armes, citoyens
Formez vos bataillons
Marchons, marchons!
Qu’un sang impur
Abreuve nos sillons!

Viva La France!

On this French National Day, the French National Anthem is sure to be played.  As you listen, remember that it is about the love of any country.  In the film Casablanca, its playing is led by a Czechoslovakian resistance leader in an American Club in French Morocco while the resistance leader’s Norwegian wife looks on and a Spanish guitarist lustily joins in.  The song is sung to drown out the singing by the Nazi soldiers.  But it is because of a love of country.  The song inspires the phrase “Long Live France”, not “Death to Germany”.  
https://www.youtube.com/watch?v=cOeFhSzoTuc

On the US's Independence Day, the Boston Pops plays the 1812 Overture which also features the melody of the French National Anthem.  But in this case it is first played triumphantly and then as a fading melody to symbolize the retreat of Napoleon’s invading Army.   This same Anthemn can symbolize love of country in the face of domination, or the failed domination by an invader.  Sing this stirring Anthem with love, and oppose domination.

Choice V

 

Freedom of Choice

A victim of collision on the open sea

Nobody ever said that life was free

Sink, swim, go down with the ship

But use your freedom of choice

But it is your choice, not my choice.

Variance, σ², is the number that expresses the range of choices.  If there are two choices, e.g. a two‑sided coin, then the two choices are 1, heads and 2, tails, and the square root of the variance, σ²,  is 1/6 because the mean choice, µ, is 1.5, and 1.5 plus 3σ, where σ is the square root of the variance, includes those two choices.  For a six-sided die, with outcomes of 1, 2, 3, 4, 5 or 6, the mean choice is 3.5, the square root of the variance is 2.5/3=.83333, and the variance is 0.69444 .  For a one-sided die, a mobius strip, where the choice is only 1, the mean choice is 0.5, which is also the odds, and the variance is 1/36.  If there is no choice, then the variance is NOT zero. It is undefined.  Acting as if there is no choice, which is virtually identical to saying that everyone should be making my choice, may be why we are in the current dilemma. 

Saying that Lies are equal to Truth means that there were two choices.  Saying that you have chosen Truth and everyone else should choose Truth does not change the fact that there were originally two choices, not one choice.  Saying that there is Truth and No Truth, but Truth is greater than No Truth means that there is only one choice,  i.e. is pro-choice, not no choice.  

The following graphs are intended to visualize the problem.  The exponential distribution is one attempt to show how outcomes are distributed given an input.  Its Probability Density Function, PDF, is λe^-λx and its Cumulative Distribution Function, CDF, is 1-e^-λx for x>0.  This can be coordinate transformed to a new location from a location of 0 to a location of μ, as PDF =λe^-λ(x­-μ) and CDF  Cumulative Distribution Function as 1-e^-λ(x-μ) , both for x>µ. However all inputs, not just x>µ , should be considered and the exponential distribution can only consider a limited number of inputs.  It might be mirrored to consider all inputs.  

A random normal distribution is the logistics distribution whose PDF is ½*(1/2s)*sech²((x-μ)/2s) and whose CDF is ½*(tanh((x-μ)/2s)+1).  S is a range variable that is related to the variance by σ²=s²π²/3.  If the variance is equal to the odds for one choice, Random Normal 2 , then its PDF looks like the exponential distribution and its mirror PDF.  However if s, not the variance, is equal to the odds for one choice, Random Normal 1, then its  PDF no longer looks like that of the PDF for the exponential distribution and its mirror.  The exponential distribution and its mirror also appears more similar to two choices rather than one choice when its CDF is considered.  And the CDF is absolutely not like no choice which would be a flat line.

Calling the variance, the odds for choice, because you don’t like the choice is no different than calling yourself Indiana because you don’t like the name Junior.  https://www.youtube.com/watch?v=DYtWWLqeSKA  It does not change the fact that your name is Henry Junior, and the dog’s name was Indiana. 

Storytelling II

 

As Time Goes By

It's still the same old story
A fight for love and glory
A case of do or die
The world will always welcome lovers
As time goes by

And we will ALWAYS need stories.

Star of stage, screen, and television? No matter what, it is all about storytelling. 

Performances/storytelling on stage are unique to a time AND place. It is a  performer's medium who decides when it is pencils down. (e.g. Taylor Swift, Hugh Jackman, Lauren Bacall, Zero Mostel, etc.)

The screen is also about storytelling, but the performance can be shown at multiple places at the same time, but only at the times dictated by those who own the screens. The screen is a director’s medium who decides when it is pencils down. (e.g. Michael Curtiz, John Ford, Steven Spielberg, Frank Capra, etc.)

Television is also storytelling but the producers decide when it is pencils down and who freeze the medium (e.g. Shonda Rimes, Quinn Martin, Norman Lear, Mark Burnett, etc.) .  The story can be told in multiple places at many times AND can be shown in the home.

Thus it is always a story, art, but who owns and sells that story, art, gets complicated.

IMHO, the art can NEVER be owned by a corporation or AI, and must be always be a human. https://dbeagan.blogspot.com/2023/06/intellectual-property.html .  IMHO, AI, is Artificial Inference,  and those inferences can NOT be made from copyrighted material, or else it is a copyright infringement. The copyright is issued by a nation, and the people of that nation own the art after the copyright has expired, and have agreed that the artist has a right to license that art for a certain period of time.

IMHO, the people also have a right to say how a copyright holder can license the art while it is copyrighted/protected. It was true when the constitution was written and it still is true. Those who create the story should be compensated for that story, or there is no incentive to create new stories.

Atheism

 

God Only Knows

God only knows what I'd be without you
God only knows what I'd be without you
If you should ever leave me
Well life would still go on believe me
The world could show nothing to me
So what good would living do me

Unless you are God, then you don’t know.

A common assumption is that all scientist are atheists.  I would suggest that nothing could be further from the truth. Scientists instead try to uncover natural laws which explain how the mind of God works.  This instead leads to a possible lack of humility that if humans can just find all of the natural laws, then humans will be equal to God and THIS can be confused with atheism.  But if the natural laws include random results, choice, and scientists are not able to predict the results of random events, then scientists have discovered how the mind of God operates, but scientists can never be equal to God.

The hubris is thus that scientists can uncover all natural laws AND that there are no random events, which is the basis for Determinism.  One of the most famous proponents of Determinism was perhaps Albert Einstein who, when confronted with the randomness of quantum mechanics, objected by supposedly saying "God does not play dice with the Universe".  In Determinism, where everything happens for a reason, if you can just discover that reason, then eliminating those reasons that cause outcomes you don’t desire will lead to only outcomes that you do desire.  If you are a believer in Determinism, then eliminating certain actions ( e.g. not stepping on cracks) or blaming actions on scapegoats, makes sense.  However if stuff just happens, then you are NOT a believer in Determinism, and superstition or scapegoating makes no sense.  Saying that God only knows and I can not ever be God is consistent with the latter position.  Saying God knows AND you can be like God is consistent with the former position.

So the question is NOT “Are scientists atheists?”.  The question is “Do Determinists believe in God, tolerance, and free will?”.

Supreme Court V

 

Opps, I Did It Again

Oops, I did it again
I played with your heart, got lost in the game
Oh baby, baby
Oops, you think I'm in love
That I'm sent from above
I'm not that innocent

Has SCOTUS did it again?

SCOTUS, especially the Robert’s court, has made some decisions with which I do not agree.  Some major recent decisions have been:

Citizens United v. FEC.  Corporations are NOT the People. The plaintiffs had no standing, and the Corporations also had no right of free speech.

Shelby County v. Holder.  Pre-clearance was being applied to a group, not to an individual.  While 50 years is indeed a long time for an individual, that time is irrelevant for a group.

Dobbs v. Jackson Women’s Health. et al. A non-viable fetus has no standing.  The taking of a woman’s womb by the state to protect a non-viable fetus for no compensation is a constitutionally prohibited taking, in this case, and in Roe v. Wade.

303 Creative LLC v. Elenis In addition to accepting lies on the part of the plaintiff, the decision also confuses the artist with the art.  In every transaction there are three components, the buyer, the seller, AND the goods and/or services being exchanged. The seller, the artist, has free speech.  The buyer, the customer, has free speech.  But the goods and services have NO such right.  I can agree that Kevin Spacey is a morally repressible human being and that Kevin Spacey deserved an Oscar for his performance in American Beauty. The two statements are not inconsistent because the artist is not his art. I can also agree that 303 Creative has free speech, and that 303 Creative can not discriminate in providing its services.

Biden v. Nebraska.  The government has a responsibility to regulate common goods.  An educated workforce is a common good.  Expenditures for an educated workforce that is directly to educational institutions, or the expenditures on the students of those institutions should be treated identically.  The forgiveness of loans after the expenses have occurred, is no different than grants to students before the expenses have occurred.  Student loan forgiveness thus appears to be a legitimate government expenditure.

Students For Fair Admissions, Inc. v. President And Fellows Of Harvard College.  The court confused the process of admitting individuals with the outcome for a group.  The actions were intended to achieve a diverse outcome for the group, and that outcome does appear to be to be of interest to the court. However the court should have made no ruling on the process by which that outcome was achieved.

In these and other cases, the Court has been improperly considering standing, has confused the rights and dominance of the individual, with certainty of the group.  Remedies to address this confusion would appear to be in order.  The court should be making decisions which increase the certainty of the Nation and protects individuals, not merely protecting the dominance of certain individuals. 

Pro-Choice

 

I’m A Man

If I had my choice of matter
I'd would rather be with cats
All engrossed in mental chatter
Showing where your mind is at

If you believe in God, then you must also believe in choice and in tolerance.

The exponential distribution is defined for the range of x>0.  It is convenient to translate from an origin of 0 to a new location, μ, and thus it becomes a range of x>μ at that new location.  It is also convenient to define x<µ as the mirror, negative of this exponential distribution.  Otherwise, when x is between 0 and µ, the probability density function and the cumulative distribution function will be undefined.

Thus a translated mirrored exponential equation also has a maximum value of 1 at the discontinuity of µ, and the combination has a Probability Density Function, PDF, and a Cummulative Density Function, CDF, of

x>μ,  PDF= λ e^{- (x-µ)} and CDF =  1-e^-λ(x-µ) 

and

x<μ, -PDF=- λ e^-λ(x-µ) and CDF = -1+e^-λ(x-µ).

While the mean of an exponential distribution is 1/λ and the median is 1/λ, the mean of the combined exponential and negative exponential distribution is 0, the median is 0 and the mode is equal to 0 but a value of 1 occurs at µ.  But this is NOT a normal distribution, becuase it is very skewed. 

The CDF of the combined exponential and negative exponential distribution appears similar to a hyperbolic tangent that has been translated to a new origin of (μ,0).  That hyperbolic tangent would have an amplitude, a, and a period, p, and thus be

a*(tanh(p*(x-µ))+C) with a constant, C, of 0, and a of 1.

which can also be expressed exponentially as

a*((e^-p*(x-µ)- e^p*(x-µ))/(e^-p*(x-µ)+ e^p*(x-µ))+C), with a constant, C,  of 0, and a of  1.

However if this is the CDF for the entire range of x, then its integral, PDF, would be 
p*sech²(p*(x‑µ)).  This is not very different than the logistics distribution, 1/(4*s)*sech²((x‑u)/(2*s)).
Its CDF is ½*tanh(((x-u)/(2*s))+ ½  which can also be expressed as an exponential as 
½*((e ^(x-µ)/2-e^-(x-µ)/2s)/(e^-(x-µ)/2s+ e^(x-µ)/2s)+ 1).  This normal CDF is a hyperbolic trigonometric function with an amplitude of ½, a period of 1/(2s) and a constant of 1. The hyperbolic tangent repeats every p/(2πi) which means that it only repeats in imagainary planes.  Because the variance, σ², for this distribution can be expressed as s²π²/3, this means that the period, p, can be expressed as √3/(2πσ).  This means that with the discontinuity, the variance must be 0, which means that the period is undefined, the amplitude must be 1, and the constant must be 0.  In other words, having no difference, the variance; no choice, the amplitude; and no God, the constant; is not normal.  Having choice, variance AND God is normal.

Independence

 

You’ll Be Back

You say the price of my love is a price you're not willing to pay
You cry in the tea which you hurled in the sea as you see me go by
Why so sad?
Remember we made an arrangement when you went away
Now you're making me mad.
Remember despite our estrangement, I'm your man

You'll be back
Soon you'll see
You'll remember you belong to me
You'll be back
Time will tell
You'll remember that I served you well
Oceans rise, empires fall
We have seen each other through it all
And when push comes to shove
I will send a fully armed battalion to remind you of my love

Was it only the taxes on Tea?

As we approach the July 4^th holiday, I am sure that most people will agree that  the phrase “No Taxation Without Representation” led to this day.  However what was not spoken and not acknowledged was the phrase “No Emancipation Without Representation” that was probably the impetus for the southern states joining in the rebellion again the King of England.  The northern states, particularly Massachusetts, as traders, were upset by the Intolerable Acts and the Tea Tax which gave the British East India Company an effective monopoly on the tea trade.  But the southern colonies did not have the type of trade that would prompt them to join with the northern trading colonies.  What prompted them to join in the rebellion? ( It is known as  the Revolution on this side of the Atlantic,  but the diffence between Rebellion and Revolution depends on the side that you support.)

The southern states were probably more upset by the 1772 case of Somerset v. Stewart. 

In that case, James Somerset, an enslaved African, was purchased by Charles Stewart, a customs officer when he was in Boston, Province of Massachusetts Bay, a British crown colony in North America.

Stewart brought Somerset with him when he returned to England in 1769, but in October 1771 Somerset escaped. After he was recaptured in November, Stewart had him imprisoned on the ship Ann and Mary (under Captain John Knowles), bound for the British colony of Jamaica. He directed that Somerset be sold to a plantation for labor. Somerset's three godparents from his baptism as a Christian in England, John Marlow, Thomas Walking and Elizabeth Cade, made an application on 3 December before the Court of King's Bench for a writ of habeas corpus. Captain Knowles on 9 December produced Somerset before the Court of King's Bench, which had to determine whether his imprisonment was lawful. https://en.wikipedia.org/wiki/Somerset_v_Stewart

Lord Mansfield, the Chief Justice of England, ruled

“The state of slavery is of such a nature that it is incapable of being introduced on any reasons, moral or political, but only by positive law, which preserves its force long after the reasons, occasions, and time itself from whence it was created, is erased from memory. It is so odious, that nothing can be suffered to support it, but positive law. Whatever inconveniences, therefore, may follow from the decision, I cannot say this case is allowed or approved by the law of England; and therefore the black must be discharged.”

The Town in Massachusetts in which I live was named for Lord Mansfield.  On this July 4^th let us remember that Independence was just as much about No Emancipation, as it was about No Taxes, without Representation.