What is the law of averages. Average values
Lecture 8. Section 1. Probability theory
Issues addressed
1) The law of large numbers.
2) Central limit theorem.
The law of large numbers.
The law of large numbers in a broad sense is understood as the general principle according to which, with a large number of random variables, their average result ceases to be random and can be predicted with a high degree of certainty.
The law of large numbers in the narrow sense means a number of mathematical theorems, in each of which, under certain conditions, the possibility of approximating the average characteristics of a large number of tests is established
to some definite constants. In the proof of theorems of this kind, the inequalities of Markov and Chebyshev are used, which are also of independent interest.
Theorem 1 (Markov inequality)... If a random variable takes non-negative values \u200b\u200band has a mathematical expectation, then for any positive number the inequality
Evidence hold for a discrete random variable. We will assume that it takes values \u200b\u200bof which the first are less or equal and all the rest are more Then
from where
Example 1. The average number of calls to the switchboard of the plant during an hour is 300. Estimate the probability that the number of calls to the switchboard within the next hour is:
1) will exceed 400;
2) there will be no more than 500.
Decision. 1) Let the random variable be the number of calls arriving at the switch during an hour. The average is. So We need to evaluate. According to Markov's inequality
2) Thus, the probability that the number of calls will be no more than 500 is not less than 0.4.
Example 2. The sum of all deposits in a bank branch is 2 million rubles, and the probability that a randomly taken deposit does not exceed 10 thousand rubles is 0.6. What about the number of contributors?
Decision. Let the randomly taken value be the size of the randomly taken contribution, and the number of all contributions. Then (thousand). According to Markov's inequality, whence
Example 3. Let be the time a student is late for a lecture, and it is known that, on average, he is late for 1 minute. Estimate the likelihood that the student will be at least 5 minutes late.
Decision.By hypothesis Applying the Markov inequality, we obtain that 
Thus, out of every 5 students, no more than 1 student will be at least 5 minutes late.
Theorem 2 (Chebyshev's inequality). 
.
Evidence. Let the random variable X be given by the distribution series
According to the definition of variance, we exclude from this sum those terms for which 
... Moreover, since all terms are non-negative, the sum can only decrease. For definiteness, we will assume that the first k terms. Then
Hence, 
.
Chebyshev's inequality allows one to estimate from above the probability of deviation of a random variable from its mathematical expectation based on information only about its variance. It is widely used, for example, in estimation theory.
Example 4. The coin is flipped 10,000 times. Estimate the probability that the incidence of the coat of arms differs from 0.01 or more.
Decision. We introduce independent random variables, where is a random variable with a distribution series
Then 
since it is distributed according to the binomial law with 
The frequency of appearance of the coat of arms is a random variable where 
... Therefore, the variance of the frequency of appearance of the coat of arms is According to the Chebyshev inequality, 
.
Thus, on average, in no more than a quarter of the cases with 10,000 coin tosses, the frequency of the coat of arms falling will differ from one hundredth or more.
Theorem 3 (Chebyshev). If are independent random variables whose variances are uniformly bounded (), then
Evidence. Because
then applying Chebyshev's inequality, we obtain
Since the probability of an event cannot be greater than 1, we obtain the required one.
Corollary 1. If are independent random variables with uniformly bounded variances and the same mathematical expectation equal to andthen
Equality (1) suggests that random deviations of individual independent random variables from their total mean value are mutually canceled out for a large mass. Therefore, although the quantities themselves are random, their mean 
when large, it is almost no longer accidental and close to. This means that if it is not known in advance, then it can be calculated using the arithmetic mean. This property of sequences of independent random variables is called the law of statistical stability. The law of statistical stability substantiates the possibility of using the analysis of statistics when making specific management decisions.
Theorem 4 (Bernoulli). If in each of p independent experiments the probability p of the occurrence of event A is constant, then
,
where is the number of occurrences of event A for these p tests.
Evidence. We introduce independent random variables, where X i - a random variable with a distribution series
Then M (X i) \u003d p, D (X i) \u003d pq. Since, then D (X i) are limited in aggregate. It follows from Chebyshev's theorem that
.
But X 1 + X 2 + ... + X P is the number of occurrences of event A in a series of p tests.
The meaning of Bernoulli's theorem lies in the fact that with an unlimited increase in the number of identical independent experiments, it can be argued with practical certainty that the frequency of occurrence of an event will differ arbitrarily little from the probability of its occurrence in a separate experiment ( statistical stability of the probability of an event). Therefore, Bernoulli's theorem serves as a bridge from application theory to its applications.
The law of large numbers
The law of the greatest numbers in probability theory states that the empirical mean (arithmetic mean) of a sufficiently large finite sample from a fixed distribution is close to the theoretical mean (mathematical expectation) of this distribution. Depending on the type of convergence, a distinction is made between the weak law of large numbers, when there is convergence in probability, and the strengthened law of large numbers, when convergence occurs almost everywhere.
There will always be such a number of tests in which, with any given in advance probability, the relative frequency of the occurrence of some event will differ as little as desired from its probability.
The general meaning of the law of large numbers is that the combined action of a large number of random factors leads to a result that is almost independent of the case.
Methods for assessing the probability based on the analysis of a finite sample are based on this property. A good example is the forecast of election results based on a survey of a sample of voters.
Weak law of large numbers
Let there be an infinite sequence (sequential enumeration) of identically distributed and uncorrelated random variables defined on the same probability space. That is, their covariance. Let be . Let's denote the sample mean of the first terms:
Stronger law of large numbers
Let there be an infinite sequence of independent identically distributed random variables defined on one probability space. Let be . Let's denote the sample mean of the first terms:
.Then almost surely.
see also
Literature
- Shiryaev A.N. Probability - M .: Science. 1989.
- Chistyakov V.P. Probability theory course, Moscow, 1982.
Wikimedia Foundation. 2010.
- Cinematography of Russia
- Gromeka, Mikhail Stepanovich
See what the "Law of Large Numbers" is in other dictionaries:
LAW OF LARGE NUMBERS - (law of large numbers) In the case when the behavior of individual members of the population is very distinctive, the behavior of the group is, on average, more predictable than the behavior of any of its members. The trend in which groups ... ... Economic Dictionary
LAW OF LARGE NUMBERS - See LARGE NUMBER LAW. Antinazi. Encyclopedia of Sociology, 2009 ... Encyclopedia of Sociology
The Law of Large Numbers - the principle according to which the quantitative laws inherent in mass social phenomena are most clearly manifested with a sufficiently large number of observations. Single phenomena are more susceptible to the influence of random and ... ... Business glossary
LAW OF LARGE NUMBERS - claims that with a probability close to one, the arithmetic mean of a large number of random variables of approximately the same order will differ little from a constant equal to the arithmetic mean of the mathematical expectations of these quantities. Various ... ... Geological encyclopedia
law of large numbers - - [Ya.N. Luginsky, M.S.Fezi Zhilinskaya, Y.S.Kabirov. English Russian Dictionary of Electrical Engineering and Power Engineering, Moscow, 1999] Subjects of electrical engineering, basic concepts EN law of averages law of large numbers ... Technical translator's guide
law of large numbers - didžiųjų skaičių dėsnis statusas T sritis fizika atitikmenys: angl. law of large numbers vok. Gesetz der großen Zahlen, n rus. the law of large numbers, m pranc. loi des grands nombres, f ... Fizikos terminų žodynas
LAW OF LARGE NUMBERS - a general principle, due to which the combined action of random factors leads, under certain very general conditions, to a result that is almost independent of the case. The convergence of the frequency of occurrence of a random event with its probability with an increase in the number ... ... Russian Sociological Encyclopedia
The law of large numbers - the law stating that the combined action of a large number of random factors leads, under some very general conditions, to a result that is almost independent of the case ... Sociology: vocabulary
LAW OF LARGE NUMBERS - statistical law expressing the relationship statistical indicators (parameters) of the sample and general population. The actual values \u200b\u200bof statistical indicators obtained for a certain sample always differ from the so-called. theoretical ... ... Sociology: Encyclopedia
LAW OF LARGE NUMBERS - the principle according to which the frequency of financial losses of a certain type can be predicted with high accuracy when there is a large number of losses of similar types ... Encyclopedic Dictionary of Economics and Law
Books
- A set of tables. Mathematics. Theory of Probability and Mathematical Statistics. 6 tables + methodology,. The tables are printed on thick polygraphic cardboard 680 x 980 mm in size. The kit includes a brochure with guidelines for teachers. Educational album of 6 sheets. Random ...
The average is the most generalized indicator in statistics. This is due to the fact that it can be used to characterize the population by quantitatively varying characteristics. For example, to compare the wages of workers of two enterprises, the wages of two specific workers cannot be taken, since it acts as a variable indicator. Also, the total amount of wages paid in enterprises cannot be taken, since it depends on the number of employees. If we divide the total wages of each enterprise by the number of employees, we can compare them and determine which enterprise has the higher average wage.
In other words, the wages of the studied set of workers receive a generalized characteristic in the average value. It expresses the general and typical that is characteristic of the aggregate of workers in relation to the trait under study. In this value, it shows the general measure of this feature, which has a different meaning for the units of the population.
Determination of the average. An average in statistics is a generalized characteristic of a set of phenomena of the same type for some quantitatively varying attribute. The average value shows the level of this trait, referred to the unit of the population. With the help of the average, it is possible to compare different populations with each other according to varying characteristics (per capita income, crop yields, production costs at various enterprises).
The average value always generalizes the quantitative variation of the trait with which we characterize the studied population, and which is equally inherent in all units of the population. This means that behind any average value there is always a series of distribution of the units of the population according to some varying feature, i.e. variation range. In this respect, the average value is fundamentally different from the relative values \u200b\u200band, in particular, from the intensity indicators. The intensity indicator is the ratio of the volumes of two different populations (for example, the production of GDP per capita), while the average summarizes the characteristics of the elements of the population according to one of the characteristics (for example, the average wage of a worker).
Average and the law of large numbers. In the change in the average indicators, a general tendency is manifested, under the influence of which the process of development of phenomena as a whole is formed, in some individual cases this tendency may not be evident. It is important that the averages are based on a mass summary of facts. Only under this condition will they reveal the general trend underlying the whole process.
In the ever more complete extinguishment of deviations generated by random causes, as the number of observations increases, the essence of the law of large numbers and its significance for mean values \u200b\u200bare manifested. That is, the law of large numbers creates the conditions for the average value to manifest a typical level of a varying feature in specific conditions of place and time. The magnitude of this level is determined by the essence of this phenomenon.
Types of averages. Averages used in statistics belong to the class of power-law means, the general formula of which is as follows:
Where x is the power average;
X - changing values \u200b\u200bof the feature (options)
- number option
Average degree indicator;
Summation sign.
With different values \u200b\u200bof the exponent of the mean, different types of mean are obtained:
Arithmetic mean;
Root mean square;
Average cubic;
Average harmonic;
Geometric mean.
Different types of average have different meanings when using the same statistical inputs. At the same time, the higher the indicator of the average degree, the higher its value.
In statistics, the correct characteristic of the population in each individual case is given only by a completely definite form of mean values. To determine this type of average value, a criterion is used that determines the properties of the average: the average value will only be the correct generalizing characteristic of the population for a varying attribute, when, when all variants are replaced with an average value, the total volume of the varying attribute remains unchanged. That is, the correct type of average is determined by how the total volume of the varying feature is formed. So, the arithmetic mean is used when the volume of a varying feature is formed as the sum of individual options, the mean square - when the volume of a varying feature is formed as the sum of squares, the harmonic mean - as the sum of the reciprocal values \u200b\u200bof individual options, the geometric mean - as the product of individual options. Apart from averages in statistics
Descriptive characteristics of the distribution of a variable attribute (structural means), fashion (the most common variant) and median (middle variant) are used.
Don't lose it. Subscribe and receive a link to the article in your mail.
Interacting daily in work or study with numbers and numbers, many of us do not even suspect that there is a very interesting law of large numbers, used, for example, in statistics, economics and even psychological and educational research. It belongs to the theory of probability and says that the arithmetic mean of any large sample from a fixed distribution is close to the mathematical expectation of this distribution.
You've probably noticed that understanding the essence of this law is not easy, especially for those who are not particularly friendly with mathematics. Based on this, we would like to talk about him simple language (as far as possible, of course), so that everyone can at least roughly understand for themselves what it is. This knowledge will help you to better understand some mathematical laws, become more erudite and influence in a positive way.
The concept of the law of large numbers and its interpretation
In addition to the above definition of the law of large numbers in the theory of probability, we can give its economic interpretation. In this case, it represents the principle that the frequency of a particular type of financial loss can be predicted with a high degree of certainty when there is high level losses of similar types in general.
In addition, depending on the level of convergence of features, we can distinguish weak and strong laws of large numbers. About the weak it comes, when convergence exists in probability, and on strengthened - when convergence exists in almost everything.
If interpreted somewhat differently, then it should be said as follows: you can always find such a finite number of tests, where with any pre-programmed probability less than one, the relative frequency of occurrence of an event will differ very little from its probability.
Thus, the general essence of the law of large numbers can be expressed as follows: the result of the complex action of a large number of identical and independent random factors will be such a result that does not depend on the case. And speaking in even simpler language, in the law of large numbers, the quantitative laws of mass phenomena will clearly manifest themselves only with a large number of them (therefore, the law is called the law of large numbers).
From this we can conclude that the essence of the law lies in the fact that in the numbers that are obtained during mass observation, there are some correctness, which cannot be found in a small number of facts.
The essence of the law of large numbers and its examples
The law of large numbers expresses the most general laws of the accidental and necessary. When random deviations "extinguish" each other, the averages determined for the same structure take the form of typical ones. They reflect the action of significant and permanent facts in specific conditions of time and place.
Regularities determined by means of the law of large numbers are strong only when they represent mass trends, and they cannot be laws for individual cases. So, the principle of mathematical statistics comes into force, which says that the complex action of a number of random factors can cause a non-random result. And the most striking example of the operation of this principle is the convergence of the frequency of occurrence of a random event and its probability when the number of tests increases.
Let's remember the usual coin toss. In theory, heads and tails can come up with the same probability. This means that if, for example, you flip a coin 10 times, 5 of them should come up tails and 5 - heads. But everyone knows that this almost never happens, because the ratio of the frequency of falling heads and tails can be 4 to 6, and 9 to 1, and 2 to 8, etc. However, with an increase in the number of coin tosses, for example, to 100, the probability of hitting heads or tails reaches 50%. If, theoretically, an infinite number of such experiments are carried out, the probability of a coin falling by both sides will always tend to 50%.
How exactly a coin falls is influenced by a huge number of random factors. This is the position of the coin in the palm of your hand, and the force with which the throw is made, and the height of the fall, and its speed, etc. But if there are many experiments, regardless of how the factors act, it can always be argued that the practical probability is close to the theoretical probability.
And here is another example that will help you understand the essence of the law of large numbers: suppose we need to estimate the level of earnings of people in a certain region. If we consider 10 observations, where 9 people receive 20 thousand rubles, and 1 person - 500 thousand rubles, the arithmetic average will be 68 thousand rubles, which, of course, is unlikely. But if we take into account 100 observations, where 99 people receive 20 thousand rubles, and 1 person - 500 thousand rubles, then when calculating the arithmetic mean we get 24.8 thousand rubles, which is closer to the real state of affairs. By increasing the number of observations, we will force the average to tend to the true indicator.
It is for this reason that in order to apply the law of large numbers, it is first of all necessary to collect statistical material in order to obtain truthful results by studying a large number of observations. That is why it is convenient to use this law, again, in statistics or social economics.
Let's sum up
The significance of the fact that the law of large numbers works is difficult to overestimate for any field of scientific knowledge, and especially for scientific developments in the field of the theory of statistics and methods of statistical knowledge. The action of the law is also of great importance for the objects under study themselves, with their mass laws. Almost all methods of statistical observation are based on the law of large numbers and the principle of mathematical statistics.
But, even without taking into account science and statistics as such, we can safely conclude that the law of large numbers is not just a phenomenon from the field of probability theory, but a phenomenon that we encounter almost every day in our life.
We hope that now the essence of the law of large numbers has become clearer to you, and you can easily and simply explain it to someone else. And if the topic of mathematics and probability theory is of interest to you in principle, then we recommend reading about and. Also get acquainted with and. And, of course, pay attention to ours, because after passing it, you will not only master new thinking techniques, but also improve your cognitive abilities in general, including mathematical ones.
