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.
http://www.buddism.ru/lib/TEXTS_/ANN/KS22.pdf
http://rain.ifmo.ru/cat/view.php/theory/processes-automata/markov-2008
http://mtkurs.ru/tipmat/kursova111.htm
http://www.exponenta.ru/educat/systemat/gomboev/labteorver/lr11/LR11.asp
etc.
Of all this, the only one that was useful to me was this - A Markov chain is a sequence of random events in which the probability of each event depends only on the state in which the process is at the current moment and is independent of earlier states.
Could you explain their meaning in simple terms.
For example, in the type of explanation and example of a Markov chain, it is one of the simplest cases of a sequence of random events. But despite its simplicity, it can often be useful even when describing rather complex phenomena.
Something about the card example is not convincing. Obviously, the order in which the cards end up after the last shuffle depends on all the shuffles before that.
If it's all about some special sense of the term "depend", then that's playing with terminology for the "chosen ones".
Could you explain its meaning in simple terms.
For example, in the type of explanation and example of a Markov chain, it is one of the simplest cases of a sequence of random events. But despite its simplicity, it can often be useful even when describing rather complex phenomena.
Alexei, could you give a clear, concise explanation of the mentioned teachings of the listed citizens, with examples.
I could, but I'm angry now. I wrote 15 lines about Bernoulli's theorem, but the forum made me re-login. It all got lost. Hang on a minute, Vladimir.
P.S. Don't even ask why the forum is so glitchy. I don't know. It's not easy to move such a big forum.
In fact, to cover the whole range of questions asked by the topicstarter, we need to write an article. For scholars. It will be very difficult, because terver/matstatistics traditionally refer to rather complicated theories: sociologists, medical workers, biologists very often very incorrectly apply terver/matstat when interpreting their observations. The reason is that their basic education is not mathematical.
In short, let's start slowly, one problem at a time.
So, here is Bernoulli's theorem in the BSE. In fact, for the humanist this article does not clarify anything, because the formulation of the theorem itself is not there. There is only an estimate of the probability of deviation of the frequency of an event from its probability (not yet confused?) by Chebyshev.
In simple, but unfortunately rather incorrect form, Bernoulli's theorem goes like this:
The frequency of an event [ in Bernoulli's scheme] tends to its probability as the number of trials increases.
To explain the formulation (especially the small print), you will have to delve at least a little bit into some basic concepts of probability theory.
1. Probability in probability theory is an indefinable concept (like straight line and point in geometry). But in order to apply it meaningfully, we need to interpret it somehow. In terversa the frequency interpretation is accepted: the probability of an event is approximately equal to the frequency of its occurrence under constant conditions of test repetition and with a very large number of tests. Let's say, if we roll the die and follow the event "Five has fallen", and our die is perfect (all faces are equally preferable), then the probability of this event p = 1/6, and the probability of the additional event ("anything has fallen but five") is q = 1 - p = 5/6. So, if we roll this die a million times, the frequency of five will be about 1/6, and the possible frequency deviations are almost always very little different from 1/6.
2. What is a Bernoulli scheme? It is a sequence of single-type and independent trials in which only 2 outcomes are possible - success (Y) and failure (F).
In our case we can take Y as the event "an A fell out" and H as "something else fell out, not equal to an A". We know the probability of success and it is p = 1/6.
The word "independent" is almost the most important thing in Bernoulli's scheme. If I'm an experienced croupier and I'm playing with someone, I can almost certainly control the game so as to turn it to my advantage. I will be able to track the results and roll the dice further so that I win. In other words, I am able to break the most important condition of trials in Bernoulli's scheme - their independence. And the probability estimates we are talking about here will be wrong.
3. We know that if we toss the die 10 times, the five can fall 0, 2, 5, and even 10 times. The most probable outcome of those mentioned is 2 times out of 10 (it is the closest to a probability of 1/6). The probability of the outcome "five never happened" is not high and not low, but for the outcome "10 out of 10 - five" it is extremely low. What laws govern these probabilities? One of the terver techniques used to find out such a law is the "multiplication" of actualizations: let's call a single sequence of 10 throws a series and now start performing many series.
If we perform many series of 10 throws (say, N = 1,000,000 series), then enter in a table the results of the series ("2 fives", "5 fives", etc.), and then draw a histogram, i.e. the dependence of frequency of series on result, we will get a curve very similar to a Gaussian one, i.e. a bell. In fact it is not a Gaussian curve, although with a million series it will differ very little from the Gaussian curve. This histogram can theoretically be calculated and it will correspond to a binomial distribution.
The main difference between the cases N=100 and N=1,000,000 will be just the "average width" of histograms. In the second case it is much smaller than the first, i.e. the histogram is narrower. "Mean width" (standard deviation) is a measure of the deviation of possible frequencies from the theoretical ones.
Now we can give voice to Bernoulli's theorem:
As the number of trials N of Bernoulli's scheme increases, the probability that the real deviation of the success rate from the probability of success will not exceed a predetermined however small epsilon>0 tends to 1.
Bernoulli's theorem does not give estimates of how large the deviation can be for a given N. These estimates can be made with the help of the Mois-Laplace theorem (local or integral). But about this - next time. For now ask questions.
P.S. I have corrected the errors in the title of the topic.
The topic is SUPER. I'm shocked by its appearance.
It's going to be tough for the authors. It's like a competent translation from Chinese.
Take your time, guys.
IMHO, it won't help. All this is empty in the absence of an appropriate base. Who has a base, he does not need to chew, those or other features to explain those or other conditions - no questions, but otherwise ... :-).
Read the primer on several occasions and YOU WILL BE RELEVANT!!! :-)
P.S. ... especially "... in plain language, without formulas." What do you mean by plain language, without formulas? One thing contradicts the other... :-) Much simpler and shorter language than having a formula! When there is a specific formula, especially with a description of its constituent variables, there is no need for any language... everything is clear.
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Could you explain its meaning in simple terms.
For example, in the type of explanation and example of a Markov chain, it is one of the simplest cases of a sequence of random events. But despite its simplicity, it can often be useful even when describing rather complex phenomena.