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.