Bernoulli, Moab-Laplace theorem; Kolmogorov criterion; Bernoulli scheme; Bayes formula; Chebyshev inequalities; Poisson distribution law; Fisher, Pearson, Student, Smirnov etc. theorems, models, simple language, without formulas. - page 5

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Let's move on. The local Moab-Laplace theorem. Picture from the same place:


The picture shows how with increasing number of trials the binomial frequency distribution tends to normal, i.e. the curve becomes more and more like a Gaussian curve (bell). And there is even a qualitative estimate of the approximation error. Thus, if we, for example, want to calculate, what is the probability that with n=200 rolls of the die m0=20 to m1=30 fives will fall (I remind that the probability of falling out of fives is 1/6), then we will not need to sum up 11 numbers with factorials, and it will be enough to calculate the corresponding area under the curve, which equation we already know. Formulas there are cumbersome, I will not give here.

Actually, in our age of personal computers this theorem is not very actual for practical computing, but 200 years ago it was quite relevant. Besides, it plays an important role in theoretical research, because the normal distribution has been studied up and down, and it is easy to work with.

Further we will talk about it, about the normal distribution, though it is not declared by the topicstarter.

 
Mathemat:

Of course, I'm not pulling, at least I'd like to make some chowder... But it's not like anyone's going to help me yet. What's a five-star cook if there's only one?

On the horizontal (abscissa) is the number of successes in the overall test series. On the vertical (ordinate) is the relative frequency, i.e. the proportion of successes in the total number of trials.

I forgot to add: binomial distribution becomes similar to normal distribution not only when n*p >= 5, but also under additional condition: p should not be too close to 1. Well, say, at p~0.5, n~10 is already quite similar.

Start by yourself and at the same time try to explain to homebrew humanitarians why they need Pearson distributions. I didn't even know they existed before you addressed me...

And explain why to express Poisson and normal (both are quite practical distributions) through the spherical horse "Pearson distribution".

But about the Gamma distribution, I'll think about it.

It's not that simple. But the Kolmogorov criterion should definitely be somewhere near the end. Chebyshev inequalities are only needed for fairly rough estimates.

Let everything remain as it is, and we will choose what we can explain on the basis of what we have learnt.

The Pearson distribution is otherwise known as the χ2 distribution. The chi-square distribution is a particular case of the gamma distribution, which http://risktheory.ru/distr_images/gammadis.gif is modelled through an exponential distribution. Values of a random variable with gamma distribution are simulated through independent realizations of exponential random variables, while values of a random variable with exponential distribution are simulated through laws and uniform distribution. Modeling values of a random variable with uniform distribution on the interval [0,1] and MO = 0.5 is available in most of modern programming systems. For example, in VBA this role is played by Rnd() function, and in Pascal and Delphi - by random function. As we can see, Gamma distribution is related to usual distributions and its origin is the usual uniform distribution and it is applied in complicated situations of this distribution, which undoubtedly includes the market, particularly Forex. Therefore, it is not accidental that all traders, sitting at the monitor screen, out of habit think they are playing with the market with 0.5 probability, but do not realize that they face a Gamma distribution, which gives them a considerably lower probability of a positive outcome. Gamma distribution can be explained to traders by means of familiar to them Fibonacci numbers, which are typical for the market due to the property that the next digit in a series is formed, considerably considering, by the sum of two previous numbers, and the Gamma function is formed, considerably considering, by the product of values of all digits in a series. Now you should feel its power, because you are already familiar with possibilities of Fibonacci levels, which are weaker than Gamma function as an integrator of properties of number series. I think that the day when Gamma levels will appear in Forex will not be far off, and maybe you will remember that this concept was first introduced in the market by yours truly.
 

I searched and found this. I see that chi-squared and gamma are special cases of Pearson distributions.

I do not see any reason to talk about Pearson distributions here, because I cannot explain the practical usefulness of such a deep vacuum-spherical horse to the readers of the branch.

I will definitely talk about chi-squared here.

Yes, perhaps we can talk about the gamut as well:

The sum of n independent exponentially distributed random variables with parameter b obeys an Erlang distribution with parameters b, n.

 
Mathemat:

I searched and found this. I see that chi-squared and gamma are special cases of Pearson distributions.

I do not see any reason to talk about Pearson distributions here, because I cannot explain the practical usefulness of such a deep vacuum-spherical horse to the readers of the branch.

I will definitely talk about chi-squared here.

Yes, perhaps we can talk about the gamut as well:

The sum of n independent exponentially distributed random variables with parameter b obeys an Erlang distribution with parameters b, n.

Now you can see in the article https://www.mql5.com/ru/articles/250 how and why this two-parameter Erlang distribution was introduced and another two-parameter distribution I introduced appeared in the body of formula (18).
 
yosuf:
Now you can see in the article https://www.mql5.com/ru/articles/250 how and why this two-parameter Erlang distribution was introduced and another two-parameter distribution I introduced appeared in the body of formula (18).

Yusuf, who were you talking to just now?
 
yosuf:
Now you can see in the article https://www.mql5.com/ru/articles/250 how and why this two-parameter Erlang distribution and another two-parameter distribution, which I introduced, were introduced in the body of formula (18).
I'll have another look. But I still do not understand how you got these probability distributions, when the article does not say anything about tervers...
 
Mathemat:
I'll have another look. I still don't understand how you got these probability distributions, when the article doesn't mention a terver...
This shows that the solutions of the material balance equations and the terver regularities coincide and they mutually complement each other when interpreting the results of the analysis of the phenomena.
 
Mathemat:

You said. There are several methods of generating a normal distribution - here, for example. But they, too, rely on a uniform distribution as a basis.

You can, of course, also "directly". We will first generate a normal distribution and then apply the function inverse of the integral function of the normal distribution to the results. But the problem is the same: it is necessary to first generate a uniform one.

Good uniform generators are described in the literature. And the last 64-bit one for Windows is not bad either, much better than the standard C-shaped one.

But the standard one is not so bad either. Anyway, the effects of its "un-naturalness" are not so easy to detect.

Natural normal - what do you need it for, S?

I don't need it. I need to feel, for those who want to understand the theorist, why the natural (non-artificial) distribution is "normal". How it turns out in nature. Understanding (feeling it in your gut) is the key to understanding 90% in the theorem. 99% of people do not feel the essence of the theories and only learn how to correctly apply formulas. For me, for example, there is no such thing as an integral and there is only a sum. Forgive me for bringing myself as an example. But in this case I'm just telling you my way of learning.
 
yosuf:
This shows that the solutions to the material balance equations and the terver law coincide and they are mutually complementary in interpreting the results of the phenomena analysis.

Yusuf, I'm sorry, but I personally am always "stressed" by the science. What does the Erlang distribution have to do with it?

Let's try one more "perception" - answer, since you are so abrasive in terms, why there are different distributions? Who registers a NEW distribution discovered by someone else? I can make up all these distributions ... a shitload of them, but no one will accept them as something new. So, what is a new distribution that is not yet known?

 

Let's listen to Alexei's presentation first, since he was the first to do so.

Yusuf and everyone else, please don't take it as a diminution of your knowledge on the subject.

This way the sequence starts to get cluttered with additional terminology and getting ahead of ourselves.

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