Beta distribution. Approximation of the law of distribution of the sum of random variables distributed according to the beta law Estimation of the parameters of the beta distribution by the number of successes

10.7256/2306-4196.2015.6.17225

Publication date:

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 an 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 of the results, indicating the appropriateness of 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 law of distribution 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 vast majority of cases of studying 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 law of distribution of the sum of beta values ​​is not only of a scientific nature, but also of a certain practical interest. At the same time, unlike most distribution laws, the beta law does not have unique properties that make it possible to analytically describe the required sum. Moreover, the specificity of this law is such that extracting a multiple definite integral, which is necessary when determining the density of the sum of random variables, is extremely difficult, and the result is a rather cumbersome expression already at n=2, and with an increase in the number of terms, the complexity of the final expression increases many times over. 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, a conclusion was made about the expediency of estimating 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 given 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, as a rule, are beta values ​​determined in an arbitrary interval . First of all, this is 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.

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 parameters p i and q i. The distribution density of individual terms will be determined by the formula:

Partially, the problem of finding the law of the sum of beta values ​​has already been solved earlier. In particular, formulas were obtained to estimate the sum of two beta values, each of which is defined using (1). An approach to finding the sum of two random variables with the distribution law (2) is proposed.

However, in the general case, the original problem has not been solved. This is primarily due to the specifics 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 desired 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, already for n=2 the use of formula (3) leads to a very cumbersome result, which is determined in terms of hypergeometric functions . Re-taking the integral of the obtained density, which must be done already at n=3 and above, is extremely difficult. At the same time, errors that will inevitably arise when rounding and calculating such a complex expression are not ruled out. In this regard, there is a need to find an approximation for formula (3), which allows using known formulas with a minimum error.

2. Computational experiment for approximating the density of the sum of beta values

To analyze the specifics of the desired distribution density, an experiment was conducted to collect statistical information about a random variable, which is the sum of a predetermined number of random variables that have a beta distribution with given parameters. The setup of the experiment was described in more detail in . By varying the parameters of individual beta values, as well as their number, as a result of a large number of experiments, we came to the following conclusions.

1. If the individual random variables included in the sum have symmetrical densities, then the histogram of the final distribution has a form close to normal. Also close to the normal law are estimates of the numerical characteristics of the final value (mathematical expectation, dispersion, asymmetry and kurtosis).

2. If individual random variables are asymmetric (both with positive and negative asymmetry), but the total asymmetry is 0, then from the point of view of graphical representation and numerical characteristics, the resulting distribution law is also close to normal.

3. In other cases, the desired 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, 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 a single law for approximation, we will carry out the following experiment. Having given the number of random variables with a beta distribution, as well as their parameters, we find the numerical value of the desired 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) for 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 approximation error by the normal distribution or beta distribution.

Let's consider these tasks in more detail. The 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 defined as follows:

`eta_(n)=xi_(1)+...+xi_(n)=eta_(n-1)+xi_(n)` , (4)

`eta_(n-1)=xi_(1)+...+xi_(n-1)` . (5)

Similarly, we 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 obtain:

`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 the beta distribution, it is first necessary to 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 compose a system of equations that includes formulas for the mathematical expectation and dispersion of the beta value:

`((Mxi=a+(b-a)p/(p+q)),(Dxi=(b-a)^(2)(pq)/((p+q)^2(p+q+1))): ) (12)`

Here `xi` is a random value that describes the desired amount. Its mathematical expectation and variance are determined by formulas (8) and (9); parameters a and b - 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 the beta values; `f_(eta)(x)` - distribution law for sum of beta values.

We will successively 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 a 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 components of the beta values;

3) how the error will change if the distribution intervals of the beta values ​​are made different.

The general scheme of the experimental algorithm for each individual values ​​of the parameters of beta values ​​can be represented as follows (Figure 2).

Figure 2 - General scheme of the experiment algorithm

PogBeta - error arising from the approximation of the final law by the beta distribution in the interval ;

PogNorm - 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 final value of the error arising from the approximation of the final distribution by the normal law.

3. Results of the experiment

Let us 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 number of terms is shown along the abscissa, and the error is shown along the ordinate. Hereinafter, the series "Norms" shows the change in the error by the normal distribution, the series "Beta" - by 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 approximation error by the beta law is approximately 4 times lower than the approximation error by the normal distribution law. Obviously, as the terms increase, the approximation error by the normal law decreases much faster than by 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 value of the error in this case, we can conclude 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 initial beta values ​​was fixed at a value of 2, and the abscissa shows the dynamics of the parameter q + 1. The y-axis on the graphs shows the approximation error. The results of the experiment with other values ​​of the parameters are generally similar.

In this case, the preference for approximating the sum of beta values ​​by the beta distribution is also obvious.

Figure 4 - Change in approximation errors with increasing asymmetry of quantities

Next, we analyzed the change in errors with a change in 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 one increases sequentially (it is plotted along the abscissa).

Figure 5 - Change in errors when changing intervals of distribution of random variables

Based on the graphical 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 the results showed, in 98% of cases the error in approximating the value under study by the beta law will be lower than in approximating the normal distribution. The average value of the approximation error beta will depend primarily on the width of the intervals over which each term is distributed. At the same time, this estimate (in contrast to the normal law) depends very little on the symmetry of random variables, as well as on the number of terms.

4. Applications

One of the areas of application of the obtained results is the problem of project management. The project is a set of mutually dependent series-parallel works with a random service duration. In this case, the duration of the project will be a random variable. Obviously, the estimation of the distribution law of this value 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 delay of the project can lead to a wide variety of adverse 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 such an 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 activities on the critical path of the project; `eta_(1)` ,..., `eta_(k)` - the duration of these jobs.

Let's consider the adjustment of the PERT method taking into account the obtained results. 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 results obtained in practice. Consider a project defined by a network diagram shown in Figure 6.

Figure 6 - Network Diagram Example

Here, the edges of the graph denote jobs, the weights of the edges denote the numbers of jobs; vertices in squares - events that signify the beginning or end of work. Let the work be given by the durations given in Table 1.

Table 1 - Time characteristics of the project work

In the above table, min is the smallest time for which this work can be completed; max - the longest time; Mat. wait. - the expectation of the beta distribution, showing the expected time to complete this work.

Let's simulate the process of project implementation using a specially developed simulation system. It is described in more detail in . As an output, you need to get:

Project histograms;

Estimation of the probabilities of project completion within 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 10000 times, we received a sample of service duration, the histogram of which is shown in Figure 7.

Figure 7 - Project Duration Histogram

Obviously, the appearance of the histogram shown in Figure 7 differs from the density plot of the normal distribution law.

We use formulas (8) and (9) to find the final mathematical expectation and variance. We get:

`Meta=27; Deta=1.3889.`

The probability of falling into 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. To do this, we use formulas (13) and (14). We get:

p=13.83; q=4.61.

The boundaries of the beta distribution will be determined by formulas (10) and (11). Will have:

The results of the study are presented in Table 2. Without loss of generality, we 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" represents the probability calculated according to the normal distribution law, which is currently used to solve the problem. The Beta column shows 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 implementing other projects, we can conclude that the estimates obtained for the approximation of the sum of random variables (2) by the beta distribution allow us to obtain a solution to this problem with greater accuracy compared to existing analogues.

The purpose of this work was to find such an approximation of the law of distribution of the sum of beta values, which would have the smallest error compared to other analogues. The following results are 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 makes it possible to obtain a numerical value of the error that occurs when the desired density is approximated 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 under various conditions. The results of the experiment 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 completion times for individual jobs. An important problem for such tasks is the assessment of risks associated with the untimely completion of the project. The results obtained make it possible to obtain more accurate estimates of the desired probabilities and, as a result, to reduce the probability of errors in planning.

Bibliography