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Khi² Tests – 03-1 – Why You Don’t Use It ?

Written By: Jean-Paul Cipria - Mar• 05•17

Do you know that one of demonstration « Khi² Distribution » cost about 29 $ ? This « Autor right » runs for 1969 [L…-1969] ! During about fifty years nobody understand correctly how to « prove » some density probabilities ! Then 99.9% of engineers don’t use it ! Is it a just return from justice ? Have we got 0.1 % to made a mistake ?


Khi2-Demonstration-2-1 – ©J.P. Cipria – 2017.


Created :2017-03-05 18:43:11. – Modified : 2017-03-13 12:55:16.

<strong><span style="color: #800000;">Very Difficult ! PhD Level.</span></strong>

Very Difficult ! PhD Level.

<strong><span style="color: #ff0000;">Expériences En Construction</span></strong>

Expériences En Construction.

Is it a good thing to bother others ?

Foreword – Fore-understanding ?

We tried since … 2014 … to explain how to … explain the khi² distribution. Some papers was written by french professors. In French High Degree School or « Ecoles Normales Supérieures » or ENS we learn to USE the khi² distribution. But … we read it, we try it, we use it. Therefore a funny then a boring sensation occurs in our brain ? Lack of « Compréhension » as the unknown word in french or no understand in english. It is a very boring sensation when we are an High Potential Student (see HP in Engineer’s Book [CIPRIA-17/10/2016]). Some post-doctorals explain us how to use it correctly but anybody seamed to explain us « How we construct the khi² density distribution ? » We asked around 200 questions with about 198 answers we understand in « Post-Master Physics in Rouen University ». Then thanks to professors, agrégés and fellows for high level pedagogy.

Then two questions remains :

  1. « How we construct the khi² density distribution ? » (And not the tautology probability solution !)
  2. « Does antenna modify Space-Time were there are arounded by an electromagnetic field ? »
    This second question as the following answer, I think :
    – « Yes, antenna modifies Space-Time referential as an magnet and its field lines. » For our point of view field line follows a trajectorie. For the magnet point of view all lines are straight lines. For example light change trajectories for our point of view and follows a straight line in the modify Space-Time. It is the same for antenna. »

Then remains : « How we construct the khi² density distribution ? » As like the « Cramer-Rao Lower Bound demonstration for a set of electromagnetic frequencies with AWG noises » we found a satisfaisant explanation after reading about 20 post-doctorates publications … without being satisfieyed. We found it somewhere else simpliest explanations … for us. Once a time … it is the story of reading ENS courses, CNRS papers and post-doc publications …

L… was money hidden and K. Pearson try to understand ?

The mathematician L… wrote a paper in 1969 [L…-1969] to explain how to prove the « khi² density distribution ». Then we can read it for 19$ for … two days or for 29$ for a pdf file. Then all others « explanations » from some mathematicians avoid to use thinking correctly to explain some evidences because some person have already a solution by « author right » ? Yes, it is !


Khi2-Demonstration-2-1 – ©J.P. Cipria – 2017.

Therefore you can read, for example, K. Pearson [PEARSON-1900]. But, as an habit, he insists on matrix results on the first part …


Khi2-Demonstration-2 – ©J.P. Cipria – 2017.

… and the correct equations were hiden in the text … ?

Khi2 - Results by K. Pearson

Khi2 – Results by K. Pearson – ©J.P. Cipria – 2017.

Mathematicians simplify to … more complexity ?

Then after a couple of years some mathematicians « arranged » equations and remarked that we can wrote it with a gamma function but … the k variable, meaning simply the number of different X random variables n, became Liberty Degree (n-1) because gamma function is shifted by one ! Then we can explain, after all, that is the number of liberties. Ok !

Matlab Program Exemple : How it work and how we can’t understand !

We copied the « simpliest » Mathematicians Khi² equation and plot it on Matlab figure. Therefore we underststood K. Pearson but, you, try to connect Person understanding and the following equation ?

%% KHI2
% -------
% JP Cipria - 28/02/2017
% La "fonction de Khi-Deux" ou encore "loi de Pearson" est un  cas particulier
% de la fonction de distribution Gamma avec k entier positif : ddl=2a : b=2
% ddl : Degré de Liberté !!! ET ALORS ???
a = ddl/2; % Fonction Khi2 ==> ddl = 2.a => si k = 1 ==> a = 0.5
% a = 1:0.1:2;
b = 2; % Fonction Khi2 ==> b = 2

% C'est la densité de probabilité de Khi2.
Khi2 = power(b,-a).*power(x,a-1).*exp(-x/b)./gamma(a);

End od Matlab program

We translate program comments issued from mathematical « explanation » we got in french courses : « The khi2 function or the Pearson law is a individual case of Gamma distribution with k >0; ddl=2a; b=2: ddl : Liberties degree !!! AND … THEN ??? »

On the same document K. Pearson explained probability p then mathématicians said q = 1 – p ! Right ! Ok. After mathematician said probability to made a mistake and K. Pearson probability to made no mistake. Ok !

Khi2-Pearson-P174 – ©J.P. Cipria - 2017.

Khi2-Pearson-P174 – ©J.P. Cipria – 2017.

As can said a french engineer : – « C’est pour emmerder le monde ? » « Is it a good thing to bother others ? »

High Potential Students drop on the dark hole ?

Then High Potential (HP) students with a very high level of understanding everythings don’t understand … everything ! And IF you are a HP and IF you don’t understand THEN you don’t use the verification method. Why ? Because we use clever-links to understand and not so « methods » or « memory ». You use transfert links to construct meanings and anticipate answers. Then if nobody explains correctly « How to Construct khi² Distribution ? » Then no engineers will use it !

Beginning of Demonstration ? Fees : 29 $ or 14 000 € ?

Fees for French Supelec Master 2 class … 14 000 €. It is why we think in … English ! 😉

What is Characteristic Functions ?

We read those things in some high level french courses but who explain you why you use characteristics functions ? What is the relationship with such a beautiful equation and the problem ?

  • \phi_x(v)=\int\limits^{\infty}_{-\infty} e^{j.uv}.p_x(u).du

How to use e^{j.uv} ? Why calculate the E(e^{j.uv}) ? The explanation is … not in french education ! It is why we think in … English !

Answer by a physicist ?

Replace V by F like frequency and U by X like a random variable. Delete all x indice ! We forgot \infty integral limits and indices like a curious physicist in 1905 ? Then …

  • \phi_(f)=\int\limits_{D} e^{j.x.f}.p(x).dx (2)

It is a Reciprocal Fourier Transform. Then we use FREQUENCY like F=1/X.

How to get probability ?

It is hidden equation like this :

  •  p(x)=\frac{1}{2.\pi}.\int\limits_{D} e^{-j.x.f}.\phi(f).df (1)

Is it an order to search solutions ?

We define the probability p(x) (1) with the frequency distribution \phi(f) NOT BY THE (2) EQUATION ! It is a tautology ! Thank you french mathematicians and Cours Préparatoires aux Grande Écoles – preparatory courses for high scientific french master’s or engineer’s schools.

But you can, you must considerate \phi(f) as a postulate or a physics law. For example Frequency Distribution is always* a physics results as a frequency density electromagnetic energy or a Bose-Einstein Density or a quantic states number [CIPRIA-25/09/2016]. And we can begin to understand ?

(*) We have got 99.9% of chance to know a physical phenomenon by is Density if you want to understand something and 0.1% to have a chance to discover first its probabilities !

Probabilité de Nombre d'États Quantiques - ©J.P. Cipria - 25/09/2016.

Probabilité de Nombre d’États Quantiques – ©J.P. Cipria – 25/09/2016.

Voir : JP. Cipria – Physique du Solide : Solid Physics : Do we search quantic states number or a pulsation (frequency ?)

… Let’s go !

Then begin to read K. Pearson in pages order and CNRS/ENS superior statistics by the … end. For HP in difficulties, don’t read the first chapter. It is the result without demonstration. Maybe prove is following ? Hope is vital ?


  • [CIPRIA-25/09/2016] : CIPRIA, Jean-Paul – « Physique du Solide – Zone de Brillouin – Probabilité d’états quantiques – Cherchons nous le nombre d’états quantiques ou une pulsation ?  » 2016.

Physique du Solide

  • [L…-1969] – L … – 1969. 29 $. No free text available then no citation.
  • [PEARSON-1900] PEARSON, Karl. « On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. » Philos. Magazine, 5th Series 50 157–175., Reprinted in Karl Pearson’s Early Statistical Papers (1948) 339–357 Cambridge Univ. Press., 1900.
  • [CIPRIA-17/10/2016] : CIPRIA, Jean-Paul – « Le contexte fait-il partie de la question ? Qu’est-ce qu’un Haut Potentiel ?
    « Are contexts integrated in the questions ? What is a High Potential Person ? » – 2016.

HP 01 – Le Contexte fait-il Partie de la Question ? Qu’est-ce qu’un HP ?


Jean-Paul Cipria

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