Consider the Beta distribution, calculate its mathematical expectation, variance, and mode. Using the MS EXCEL BETA.DIST () function, we will plot the graphs of the distribution function and probability density. Let's generate an array of random numbers and evaluate the distribution parameters.
Beta distributionBeta-
distribution)
depends on 2 parameters: α ( alpha)> 0(determines the shape of the distribution) and b (beta)> 0(determines the scale).
Unlike many other continuous distributions, the range of variation of a random variable having Beta distribution, is limited by the segment. Outside this segment distribution density equals 0. The boundaries of this segment are set by the researcher depending on the problem. If A = 0 and B = 1, then such Beta distribution called standard.
Beta distribution has the designation Beta(alpha; beta).
Note: If the parameters alpha and beta= 1, then Beta distribution turns into, i.e. Beta (1; 1; A; B) = U (A; B).
In general distribution function cannot be expressed in elementary functions, therefore it is calculated by numerical methods, for example, using the MS EXCEL BETA.DIST () function.
Note: For the convenience of writing formulas in the example file for the distribution parameters alpha and beta appropriate.
The example file also contains graphs probability density and distribution functions with marked values middle, and .
Random number generation and parameter estimation
Using inverse distribution function(or quantile values ( p-
quantile),
see) you can generate values of a random variable having Beta distribution... To do this, you need to use the formula:
BETA.OBR (RAND (); alpha; beta; A; B)
ADVICE: Because random numbers are generated using the RAND () function, then pressing the key F9, it is possible to obtain a new sample each time and, accordingly, a new estimate of the parameters.
The RAND () function generates from 0 to 1, which exactly corresponds to the range of variation of the probability (see. example file sheet Generation).
Now having an array of random numbers generated with the given distribution parameters alpha and beta(let there be 200), let us estimate the distribution parameters.
Parameter estimation alpha and beta can be done with method of moments(it is assumed that parameters A and B are known):
Correct link to this article:
Oleinikova S.A. - Approximation of the distribution law of the sum of random variables distributed according to the beta law // Cybernetics and programming. - 2015. - No. 6. - P. 35 - 54. DOI: 10.7256 / 2306-4196.2015.6.17225 URL: https://nbpublish.com/library_read_article.php?id=17225
Approximation of the distribution law of the sum of random variables distributed according to the beta law
| Oleinikova Svetlana Alexandrovna Doctor of Technical Sciences Associate Professor, Voronezh State Technical University 394026, Russia, Voronezh, Moskovsky prospect, 14 Oleinikova Svetlana Aleksandrovna Doctor of Technical Science Associate Professor, Department of Automated and Computing Systems, Voronezh State Technical University 394026, Russia, g. Voronezh, Moskovskii prospekt, 14 Date of sending the article to the editor: 14-12-2015
Date of review of the article: 15-12-2015
Annotation. The subject of research in this work is the distribution density of a random variable, which is the sum of a finite number of beta values, each of which is distributed in its own interval with its own parameters. This law is widespread in probability theory and mathematical statistics, since it can be used to describe a sufficiently large number of random phenomena if the values of the corresponding continuous random variable are concentrated in a certain interval. Since the sought sum of beta values cannot be expressed by any of the known laws, the problem arises of estimating its distribution density. The aim of the work is to find such an approximation for the distribution density of the sum of beta values, which would differ in the smallest error. To achieve this goal, a computational experiment was carried out, as a result of which, for a given number of beta values, the numerical value of the distribution density was compared with the approximation of the desired density. Normal and beta distributions were used as approximations. As a result of the experimental analysis, results were obtained that indicate the advisability of approximating the sought distribution law by the beta law. As one of the areas of application of the results obtained, the problem of project management with a random duration is considered, where the key role is played by the estimation of the project execution time, which, due to the specifics of the subject area, can be described using the sum of beta values.
Keywords:
random variable, beta distribution, distribution density, normal distribution law, sum of random variables, computational experiment, recursive algorithm, approximation, error, PERT
10.7256/2306-4196.2015.6.17225
Date of publication: 19-01-2016
Abstract. The subject of the research in this paper is the probability density function (PDF) of the random variable, which is the sum of a finite number of beta values. This law is widespread in the theory of probability and mathematical statistics, because using it can be described by a sufficiently large number of random events, if the value of the corresponding continuous random variable concentrated in a certain range. Since the required sum of beta values can not be expressed by any of the known laws, there is the problem of estimating its density distribution. The aim is to find such approximation for the PDF of the sum of beta-values that would have the least error. To achieve this goal computational experiment was conducted, in which for a given number of beta values the numerical value of the PDF with the approximation of the desired density were compared. As the approximations it were used the normal and the beta distributions. As a conclusion of the experimental analysis the results, indicating the appropriateness the approximation of the desired law with the help of the beta distribution, were obtained. As one of the fields of application of the results the project management problem with the random durations of works is considered. Here, the key issue is the evaluation of project implementation time, which, because of the specific subject area, can be described by the sum of the beta values.
Keywords: Random value, beta distribution, density function, normal distribution, the sum of random variables, computational experiment, recursive algorithm, approximation, error, PERT Introduction The problem of estimating the distribution law of the sum of beta-values is considered. This is a universal law that can be used to describe most random phenomena with a continuous distribution law. In particular, in the overwhelming number of cases of investigating random phenomena that can be described by unimodal continuous random variables lying in a certain range of values, such a value can be approximated by the beta law. In this regard, the problem of finding the distribution law for the sum of beta-values is not only scientific in nature, but also of certain practical interest. Moreover, unlike most distribution laws, the beta law does not have unique properties that allow an analytical description of the desired amount. Moreover, the specificity of this law is such that it is extremely difficult to extract a multiple definite integral necessary for determining the density of a sum of random variables, and the result is a rather cumbersome expression even for n = 2, and with an increase in the number of terms, the complexity of the final expression increases many times. In this regard, the problem arises of approximating the distribution density of the sum of beta values with a minimum error.
This paper presents an approach to finding an approximation for the desired law by means of a computational experiment that allows for each specific case to compare the error obtained by estimating the density of interest using the most appropriate laws: normal and beta. As a result, it was concluded that it is advisable to estimate the sum of beta values using the beta distribution. 1. Statement of the problem and its features In general, the beta law is determined by the density specified in the interval as follows:
`f_ (xi_ (i)) (x) = ((0,; t<0), ((t^(p_(i)-1)(1-t)^(q_(i)-1))/(B(p_(i),q_(i))(b_(i)-a_(i))^(p_(i)+q_(i)-1)), ; 0<=t<=1;),(0, ; t>1):} (1)`
However, of practical interest are, as a rule, beta values determined in an arbitrary interval. This is primarily due to the fact that the range of practical problems in this case is much wider, and, secondly, when finding a solution for a more general case, it will not be possible to obtain a result for a particular case, which will be determined by a random variable (1). present no difficulty. Therefore, in what follows we will consider random variables defined on an arbitrary interval. In this case, the problem can be formulated as follows.
We consider the problem of estimating the distribution law of a random variable, which is the sum of random variables `xi_ (i),` i = 1, ..., n, each of which is distributed according to the beta law in the interval with the parameters p i and q i. The distribution density of individual terms will be determined by the formula:
The problem of finding the law of the sum of beta values has been partially solved earlier. In particular, formulas were obtained to estimate the sum of two beta values, each of which is determined using (1). In the proposed approach to the search for the sum of two random variables with the distribution law (2).
However, in the general case, the original problem has not been solved. This is primarily due to the specificity of formula (2), which does not allow one to obtain compact and convenient formulas for finding the density from the sum of random variables. Indeed, for two quantities `xi_1` and` xi_2` the required density will be determined as follows:
`f_ (eta) (z) = int_-prop ^ propf_ (xi_1) (x) f_ (xi_2) (z-x) dx (3)`
In the case of adding n random variables, a multiple integral is obtained. At the same time, for this problem there are difficulties associated with the specifics of the beta distribution. In particular, even for n = 2, the use of formula (3) leads to a rather cumbersome result, which is defined in terms of hypergeometric functions. Re-taking the integral of the obtained density, which must be done already at n = 3 and higher, is extremely difficult. At the same time, errors are not excluded that will inevitably arise when rounding and calculating such a complex expression. In this regard, it becomes necessary to search for an approximation for formula (3), which makes it possible to apply well-known formulas with a minimum error.
2. Computational experiment to approximate the density of the sum of beta values To analyze the specifics of the desired distribution density, an experiment was carried out that allows collecting statistical information about a random variable, which is the sum of a predetermined number of random variables with a beta distribution with given parameters. The experimental setup was described in more detail in. Varying the parameters of individual beta values, as well as their number, as a result of a large number of experiments carried out, we came to the following conclusions.
1. If individual random variables included in the sum have symmetric densities, then the histogram of the final distribution has a form close to normal. They are also close to the normal law of evaluating the numerical characteristics of the final value (mathematical expectation, variance, asymmetry and kurtosis).
2. If individual random variables are asymmetric (with both positive and negative asymmetries), but the total asymmetry is 0, then from the point of view of graphical representation and numerical characteristics, the obtained distribution law is also close to normal.
3. In other cases, the sought law is visually close to the beta law. In particular, the sum of five asymmetric random variables is shown in Figure 1.
Figure 1 - The sum of five equally asymmetric random variables
Thus, on the basis of the experiment carried out, it is possible to put forward a hypothesis about a possible approximation of the density of the sum of beta values by a normal or beta distribution.
To confirm this hypothesis and choose the only law for the approximation, we will carry out the following experiment. Having given the number of random variables with beta distribution, as well as their parameters, we find the numerical value of the required density and compare it with the density of the corresponding normal or beta distribution. This will require:
1) develop an algorithm that allows you to numerically estimate the density of the sum of beta values;
2) with the given parameters and the number of initial values, determine the parameters of the final distribution under the assumption of a normal or beta distribution;
3) determine the error of approximation by the normal distribution or the beta distribution.
Let's consider these tasks in more detail. A numerical algorithm for finding the density of the sum of beta values is based on recursion. The sum of n arbitrary random variables can be determined as follows:
`eta_ (n) = xi_ (1) + ... + xi_ (n) = eta_ (n-1) + xi_ (n)` , (4)
`eta_ (n-1) = xi_ (1) + ... + xi_ (n-1)` . (5)
Similarly, you can describe the distribution density of the random variable `eta_ (n-1)`:
`eta_ (n-1) = xi_ (1) + ... + xi_ (n-1) = eta_ (n-2) + xi_ (n-1)` , (6)
Continuing similar reasoning and using formula (3), we get:
`f_ (eta_ (n)) (x) = int_-prop ^ prop (f_ (xi_ (n-1)) (x-x_ (n-1)) * int_-prop ^ prop (f_ (xi_ (n- 2)) (x_ (n-1) -x_ (n-2)) ... int_-prop ^ propf_ (xi_ (2)) (x_ (2) -x_ (1)) dx_ (1) ... ) dx_ (n-2)) dx_ (n-1). (7) `
These considerations, as well as the specifics of determining the density for quantities with a beta distribution, are given in more detail in.
The parameters of the final distribution law are determined based on the assumption of the independence of random variables. In this case, the mathematical expectation and variance of their sum will be determined by the formulas:
`Meta_ (n) = Mxi_ (1) + ... + Mxi_ (n), (8)`
For the normal law, the parameters a and `sigma` will be directly determined by formulas (8) and (9). For beta distribution, you must first calculate the lower and upper bounds. They can be defined as follows:` `
`a = sum_ (i = 1) ^ na_ (i)`; (ten)
,,, b = sum_ (i = 1) ^ nb_ (i) `. (eleven)
Here a i and b i are the boundaries of the intervals of individual terms. Next, we will compose a system of equations that include formulas for the mathematical expectation and variance of the beta value:
`((Mxi = a + (ba) p / (p + q)), (Dxi = (ba) ^ (2) (pq) / ((p + q) ^ 2 (p + q + 1))): ) (12) `
Here `xi` is a random variable describing the required sum. Its mathematical expectation and variance are determined by formulas (8) and (9); parameters a and b are given by formulas (10) and (11). Having solved system (12) with respect to the parameters p and q, we will have:
`p = ((b-Mxi) (Mxi-a) ^ 2-Dxi (Mxi-a)) / (Dxi (b-a))` . (13)
`q = ((b-Mxi) ^ 2 (Mxi-a) -Dxi (b-Mxi)) / (Dxi (b-a))` . (14)
`E = int_a ^ b | hatf (x) -f_ (eta) (x) | dx. (15) `
Here `hatf (x)` is an approximation of the sum of beta values; `f_ (eta) (x)` - distribution law of the sum of beta values.
We will sequentially change the parameters of individual beta values to estimate the errors. In particular, the following questions will be of interest:
1) how quickly the sum of beta values converges to the normal distribution, and is it possible to estimate the sum by another law that will have a minimum error relative to the true distribution law of the sum of beta values;
2) how much the error increases with an increase in the asymmetry of the beta-values;
3) how the error will change if the distribution intervals of beta values are made different.
The general scheme of the experiment algorithm for each individual values of the beta-values can be represented as follows (Figure 2).
Figure 2 - General scheme of the experiment algorithm
PogBeta - the error arising from the approximation of the final law by the beta distribution in the interval;
PogNorm - the error arising from the approximation of the final law by a normal distribution in the interval;
ItogBeta - the final value of the error arising from the approximation of the final distribution by the beta law;
ItogNorm - the total value of the error arising from the approximation of the final distribution by the normal law. 3. Experimental results Let's analyze the results of the experiment described earlier.
The dynamics of the decrease in errors with an increase in the number of terms is shown in Figure 3. The abscissa shows the number of terms, and the ordinate shows the magnitude of the error. Hereinafter, the "Norm" series shows the change in the error by the normal distribution, the "Beta" series - the beta - distribution.
Figure 3 - Reduction of errors with a decrease in the number of terms
As can be seen from this figure, for two terms, the error of approximation by the beta law is about 4 times lower than the error of approximation by the normal distribution law. Obviously, as the terms increase, the approximation error by the normal law decreases much faster than the beta law. It can also be assumed that for a very large number of terms, the approximation by the normal law will have a smaller error than the approximation by the beta distribution. However, taking into account the magnitude of the error in this case, it can be concluded that from the point of view of the number of terms, the beta distribution is preferable.
Figure 4 shows the dynamics of changes in errors with an increase in the asymmetry of random variables. Without loss of generality, the parameter p of all the initial beta values was fixed with a value of 2, and the dynamics of the change in the parameter q + 1 is shown on the abscissa axis. The ordinate axis in the graphs shows the approximation error. The results of the experiment with other values of the parameters are generally similar.
In this case, it is also obvious that it is preferable to approximate the sum of beta values by a beta distribution.
Figure 4 - Change in approximation errors with increasing asymmetry of quantities
Next, we analyzed the change in errors when changing the range of the initial beta values. Figure 5 shows the results of measuring the error for the sum of four beta values, three of which are distributed in the interval, and the range of the fourth increases sequentially (it is plotted on the abscissa).
Figure 5 - Change in errors when changing the intervals of distribution of random variables
Based on the graphic illustrations shown in Figures 3-5, as well as taking into account the data obtained as a result of the experiment, it can be concluded that it is advisable to use the beta distribution to approximate the sum of beta values.
As shown by the results obtained, in 98% of cases, the error in approximating the investigated value by the beta law will be lower than in approximating the normal distribution. The average value of the beta approximation error will depend primarily on the width of the intervals over which each term is distributed. In this case, this estimate (in contrast to the normal law) depends very little on the symmetry of the random variables, as well as on the number of terms. 4. Applications One of the areas of application of the results obtained is the task of project management. A project is a set of mutually dependent serial-parallel jobs with a random service duration. In this case, the duration of the project will be a random value. Obviously, the assessment of the distribution law of this quantity is of interest not only at the planning stages, but also in the analysis of possible situations associated with the untimely completion of all work. Taking into account the fact that the project delay can lead to a wide variety of unfavorable situations, including fines, the estimation of the distribution law of a random variable describing the duration of the project seems to be an extremely important practical task.
Currently, the PERT method is used for this assessment. According to his assumptions, the duration of the project is a normally distributed random variable `eta` with parameters:
`a = sum_ (i = 1) ^ k Meta_ (i)`, (16)
`sigma = sqrt (sum_ (i = 1) ^ k D eta_ (i))` . (17)
Here k is the number of jobs on the critical path of the project; `eta_ (1)`, ..., `eta_ (k)` - duration of these works.
Let's consider the correction of the PERT method, taking into account the results obtained. In this case, we will assume that the duration of the project is distributed according to the beta law with parameters (13) and (14).
Let's try the obtained results in practice. Consider a project defined by the network diagram shown in Figure 6.
Figure 6 - Network diagram example
Here, the edges of the graph indicate the jobs, the weights of the edges indicate the numbers of the jobs; vertices in squares - events that signify the beginning or end of work. Let the works be given by the durations given in Table 1.
Table 1 - Time characteristics of project works
| Work no. |
min |
max |
Mat. standby |
| 1
|
5
|
10
|
9
|
| 2
|
3
|
6
|
4
|
| 3
|
6
|
8
|
7
|
| 4
|
4
|
7
|
6
|
| 5
|
4
|
7
|
7
|
| 6
|
2
|
5
|
3
|
| 7
|
4
|
8
|
6
|
| 8
|
4
|
6
|
5
|
| 9
|
6
|
8
|
7
|
| 10
|
2
|
6
|
4
|
| 11
|
9
|
13
|
12
|
| 12
|
2
|
6
|
3
|
| 13
|
5
|
7
|
6
|
In the above table, min is the shortest time in which this work can be completed; max - longest time; Mat. standby is the mathematical expectation of the beta distribution, showing the expected time to complete a given job.
We will simulate the project execution process using a specially developed simulation modeling system. It is described in more detail in. As the output, you need to get:
Project histograms;
Evaluation of the probabilities of project execution in a given interval based on the statistical data of the simulation system;
Estimation of probabilities using normal and beta distributions.
During the simulation of the project execution 10,000 times, a sample of the service duration was obtained, the histogram of which is shown in Figure 7.
Figure 7 - Project duration histogram
It is obvious that the appearance of the histogram shown in Figure 7 differs from the density graph of the normal distribution law.
We will use formulas (8) and (9) to find the final mathematical expectation and variance. We get:
`M eta = 27; D eta = 1.3889.`
The probability of hitting a given interval will be calculated using the well-known formula:
`P (l (18)
where `f_ (eta) (x)` is the distribution law of the random variable `eta`, l and r- the boundaries of the interval of interest.
Let's calculate the parameters for the final beta distribution. For this we use formulas (13) and (14). We get:
p = 13.83; q = 4.61.
The boundaries of the beta distribution are determined by formulas (10) and (11). Will have:
The results of the study are given in Table 2. Without loss of generality, let us choose the number of model runs equal to 10000. In the "Statistics" column, the probability obtained on the basis of statistical data is calculated. The column "Normal" shows the probability calculated according to the normal distribution law, which is now used to solve the problem. The Beta column contains the probability value calculated from the beta distribution.
Table 2 - Results of probabilistic estimates
Based on the results presented in Table 2, as well as similar results obtained in the course of modeling the process of performing other projects, it can be concluded that the obtained estimates of the approximation of the sum of random variables (2) by the beta distribution make it possible to obtain a solution to this problem with greater accuracy compared to existing counterparts. The aim of this work was to find such an approximation of the distribution law of the sum of beta values, which would differ in the smallest error in comparison with other analogs. The following results were obtained.
1. Experimentally, a hypothesis was put forward about the possibility of approximating the sum of beta values using the beta distribution.
2. A software tool has been developed that allows one to obtain the numerical value of the error arising from the approximation of the desired density by the normal distribution law and the beta law. This program is based on a recursive algorithm that allows you to numerically determine the density of the sum of beta values with a given density, which is described in more detail in.
3. A computational experiment was set up, the purpose of which was to determine the best approximation by comparative analysis of errors in different conditions. The experimental results showed the feasibility of using the beta distribution as the best approximation of the distribution density of the sum of beta values.
4. An example is presented in which the results obtained are of practical importance. These are project management tasks with random execution times for individual jobs. An important problem for such tasks is the assessment of the risks associated with the late completion of the project. The results obtained make it possible to obtain more accurate estimates of the desired probabilities and, as a consequence, to reduce the probability of errors in planning.
Bibliography
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Closed educational course for children of the elite: "The true arrangement of the world."
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From Wikipedia, the free encyclopedia
Beta distribution
Probability density
Probability density function for the Beta distribution |
Distribution function
Cumulative distribution function for the Beta distribution |
| Designation
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texvc not found; See math / README for setup help.): \ Text (Be) (\ alpha, \ beta)
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| Options
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Unable to parse expression (Executable texvc not found; See math / README - tuning reference.): \ Alpha> 0
Unable to parse expression (Executable texvc not found; See math / README for configuration help.): \ Beta> 0
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| Carrier
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Unable to parse expression (Executable texvc not found; See math / README for configuration help.): X \ in
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| Probability density
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Unable to parse expression (Executable texvc not found; See math / README for setup help.): \ Frac (x ^ (\ alpha-1) (1-x) ^ (\ beta-1)) (\ mathrm (B) (\ alpha, \ beta))
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| Distribution function
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Unable to parse expression (Executable texvc not found; See math / README for configuration help.): I_x (\ alpha, \ beta)
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| Expected value
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Unable to parse expression (Executable texvc not found; See math / README for tuning help.): \ Frac (\ alpha) (\ alpha + \ beta)
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| Median
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Unable to parse expression (Executable texvc not found; See math / README for tuning help.): \ Frac (\ alpha-1) (\ alpha + \ beta-2) for Unable to parse expression (Executable texvc not found; See math / README for tuning help.): \ Alpha> 1, \ beta> 1
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| Dispersion
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Unable to parse expression (Executable texvc not found; See math / README for setup help.): \ Frac (\ alpha \ beta) ((\ alpha + \ beta) ^ 2 (\ alpha + \ beta + 1))
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| Asymmetry coefficient
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Unable to parse expression (Executable texvc not found; See math / README for setup help.): \ Frac (2 \, (\ beta- \ alpha) \ sqrt (\ alpha + \ beta + 1)) ((\ alpha + \ beta + 2) \ sqrt (\ alpha \ beta))
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| Kurtosis coefficient
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Unable to parse expression (Executable texvc not found; See math / README for setup help.): 6 \, \ frac (\ alpha ^ 3- \ alpha ^ 2 (2 \ beta-1) + \ beta ^ 2 (\ beta + 1) -2 \ alpha \ beta (\ beta + 2)) (\ alpha \ beta (\ alpha + \ beta + 2) (\ alpha + \ beta + 3))
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| Differential entropy
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| Generating function of moments
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Unable to parse expression (Executable texvc not found; See math / README for setup help.): 1 + \ sum_ (k = 1) ^ (\ infty) \ left (\ prod_ (r = 0) ^ (k-1) \ frac (\ alpha + r) (\ alpha + \ beta + r) \ right) \ frac (t ^ k) (k
!}
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| Characteristic function
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Unable to parse expression (Executable texvc not found; See math / README for setup help.): () _1F_1 (\ alpha; \ alpha + \ beta; i \, t)
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Beta distribution in probability theory and statistics, a two-parameter family of absolutely continuous distributions. Used to describe random variables whose values are limited to a finite interval.
Definition
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Probability distributions |
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Multidimensional |
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Bernoulli | Binomial | Geometric | Hypergeometric | Logarithmic | Negative binomial | Poisson | Discrete uniform |
Multinomial |
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| Absolutely continuous:
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Beta| Weibulla | Gamma | Hyperexponential | Gompertz Distribution | Kolmogorov | Cauchy | Laplace | Lognormal | | | Kopula |
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An excerpt characterizing the Beta distribution
Tears glistened in my eyes ... And I was not at all ashamed of it. I would give a lot to meet one of them alive! .. Especially Magdalena. What wondrous, ancient Magic burned in the soul of this amazing woman when she created her magical kingdom ?! The kingdom in which Knowledge and Understanding ruled, and the backbone of which was Love. Only not the love that the “holy” church shouted about, having worn out this wondrous word to the point that I did not want to hear it any longer, but that beautiful and pure, real and courageous, the only and amazing LOVE with whose name the powers were born ... and with whose name the ancient warriors rushed into battle ... with whose name a new life was born ... by whose name our world changed and became better ... This Love was carried by the Golden Mary. And it is to this Mary that I would like to bow ... For everything that she carried, for her pure, bright LIFE, for her courage and courage, and for Love.
But, unfortunately, it was impossible to do this ... She lived centuries ago. And I couldn't be the one who knew her. An incredibly deep, light sadness suddenly swept over my head, and bitter tears poured down ...
- Well, what are you, my friend! .. Other sorrows await you! - Sever exclaimed in surprise. - Please, calm down ...
He gently touched my hand and gradually the sadness disappeared. Only bitterness remained, as if I had lost something light and expensive ...
- You cannot relax ... War awaits you, Isidora.
- Tell me, Sever, was the teaching of the Cathars called the Teaching of Love because of the Magdalene?
- Here you are not quite right, Isidora. The uninitiated called him the Teaching of Love. For those who understood, it carried a completely different meaning. Listen to the sound of the words, Isidora: love in French sounds - amor - isn't it? And now, strip this word, separating the letter "a" from it ... It will turn out a'mor (a "mort) - without death ... This is the true meaning of the Magdalene's teachings - the Teaching of the Immortals. As I told you before - everything simply, Isidora, if only to look and listen correctly ... Well, and for those who do not hear - let it remain the Teaching of Love ... it is beautiful too.
I stood completely dumbfounded. The Teaching of the Immortals! .. Daariya ... So, what was the teaching of Radomir and Magdalene! .. The North surprised me many times, but never before I felt so shocked! .. The Teaching of the Cathars attracted me with its powerful, magical power, and I could not forgive myself for not talking about this with the North before.
- Tell me, Sever, is there anything left of the Qatar records? Something must have survived, right? Even if not the Perfect ones themselves, then at least just disciples? I mean something about their real life and teaching?
- Unfortunately - no, Isidora. The Inquisition destroyed everything, everywhere. Her vassals, by order of the Pope, were even sent to other countries to destroy every manuscript, every remaining piece of birch bark that they could find ... We were looking for at least something, but we could not save anything.
- Well, what about the people themselves? Couldn't there be something left for people who would keep it through the centuries?
- I don’t know, Isidora ... I think, even if someone had some kind of recording, it was changed over time. After all, it is natural for a person to reshape everything in his own way ... And especially without understanding. So it is unlikely that anything has survived as it was. It's a pity ... True, we have preserved the diaries of Radomir and Magdalene, but that was before the creation of the katar. Although, I think, the teaching has not changed.
- Sorry, for my confused thoughts and questions, Sever. I see that I have lost a lot without coming to you. But still, I'm still alive. And while I'm breathing, I can still ask you, can't I? Can you tell me how Svetodar's life ended? Sorry to interrupt.
Sever smiled sincerely. He liked my impatience and my thirst to "have time" to find out. And he continued with pleasure.
After his return, Svetodar lived and taught in Occitania for only two years, Isidora. But these years became the most expensive and happiest years of his wandering life. His days, illuminated by Beloyar's cheerful laughter, passed in his beloved Montsegur, surrounded by the Perfect ones, to whom Svetodar honestly and sincerely tried to convey what the distant Wanderer had taught him for many years.
- Bernoulli formula.
Itself distribution
are called binomial.
The parameters of the binomial distribution are the probability of success p (q = 1 - p) and the number of trials n. The binomial distribution is useful for describing the distribution of binomial events, such as the number of men and women in randomly selected companies. The use of the binomial distribution in game problems is of particular importance.
The exact formula for the probability m of successes in n trials is written as follows:
where p is the probability of success; q is 1-p, q> = 0, p + q = 1; n - number of tests, m = 0.1 ... m
The main characteristics of the binomial distribution:
6. Poisson's formula and Poisson distribution.
Let the number of trials n be large, the probability p small, and
np is small. Then the probability of m successes in n trials can be approximately determined by Poisson's formula:
.
A random variable with a distribution series m,
has a Poisson distribution. The more n, the more accurate the Poisson formula. For rough calculations, the formula is used for n = 10,
0 - 2, for n = 100
0 - 3. In engineering calculations, the formula is applied when n = 20,
0 - 3, n = 100,
0 - 7. For accurate calculations, the formula is applied when n = 100,
0 - 7, n = 1000,
0
– 15.
Let us calculate the mathematical expectation and variance of a random variable with a Poisson distribution.
The main characteristics of a Poisson random variable:
Poisson Distribution Plot:
7. Geometric distribution.
Consider the Bernoulli scheme. Let's designate X - the number of trials before the first success, if the probability of success in one trial is p. If the first test is successful, then X = 0. Therefore,
... If X = 1, i.e. the first test is unsuccessful, and the second is successful, then by the multiplication theorem
... Similarly, if X = n, then all tests up to the n-th test are unsuccessful and
... Let's compose a series of distribution of a random variable X
A random variable with such a distribution series has geometric distribution.
Let us check the normalization condition:
8. Hypergeometric distribution.
This is a discrete probability distribution of a random variable X taking integer values m = 0, 1,2, ..., n with probabilities:
where N, M and n are non-negative integers and M< N, n < N.
The mathematical expectation of the hypergeometric distribution does not depend on N and coincides with the mathematical expectation µ = np of the corresponding binomial distribution.
Dispersion of the hypergeometric distribution 
does not exceed the variance of the binomial distribution npq. Instances of any order of the hypergeometric distribution tend to the corresponding values of the moments of the binomial distribution.
9. Beta distribution.
The beta distribution has a density of the form:
The standard beta distribution is concentrated in the range from 0 to 1. Applying linear transformations, the beta value can be transformed so that it will take values at any range.
The main numerical characteristics of a quantity with a beta distribution:
Noun., Number of synonyms: 1 distribution (62) ASIS synonym dictionary. V.N. Trishin. 2013 ... Synonym dictionary
beta distribution- 1.45. beta distribution Probability distribution of a continuous random variable X, which can take any values from 0 to 1, including boundaries, and whose distribution density at 0 £ x £ 1 and parameters m1> 0, m2> 0, where Г ... ... Dictionary-reference book of terms of normative and technical documentation
beta distribution- Probability distribution of a continuous random variable taking values on a segment, the density of which is given by the formula, where, a, b> 0 and is the gamma function. Note. Its special cases are many widely used ... ... Dictionary of Sociological Statistics
See plan ... Synonym dictionary
In probability theory and mathematical statistics, the Dirichlet distribution (named after Johann Peter Gustave Lejeune Dirichlet) often denoted Dir (α) is a family of continuous multivariate probability distributions parameterized by the vector α ... ... Wikipedia
Beta: Wiktionary has an entry "beta" Beta (letter) (β) is the second letter of the Greek alphabet. Beta testing Beta coefficient Beta function (mathematics) Beta distribution (probability theory ... Wikipedia
Probability density ... Wikipedia
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- Comparison of admission to educational programs at a university based on the results of Olympiads and USE scores, O. V. Poldin. In the article, to compare the quality of admission to universities for various educational programs, it is proposed to use the adjusted demand curves obtained from the results of the USE of those enrolled in ...
