Some random events are irregular, but many are regular. Bernoulli, a mathematician of these "regular random events", often presents almost inevitable statistical characteristics in the case of a large number of repeated occurrences. This law is the law of large numbers. To be exact, the law of large numbers expresses the statistical regularity of a large number of repeated random phenomena in an accurate mathematical form, that is, the stability of frequency and the stability of average results, and discusses their conditions. [2]
In layman's terms, this theorem is that when the experiment is repeated many times, the frequency of random events approaches its probability. For example, when we toss a coin up, which side of the coin falls by accident, but when we toss a coin up enough times, reaching tens of thousands or even hundreds of millions of times, we will find that each side of the coin accounts for about half of the total times. In this case, contingency contains inevitability. The inevitable laws and characteristics are reflected in a large number of samples.
The theorem of large numbers is simply that "when the number of experiments is enough, the frequency of events is infinitely close to the probability of events". This description is Bernoulli's law of large numbers.
The significance of this law is that when n is large, the arithmetic mean of random variables subject to the same distribution will approach the mathematical expectation of these random variables in probability.
Applying this rule to sampling survey, we can draw the following conclusions: With the increase of sample size n, the sample mean will approach the overall mean. This provides a theoretical basis for estimating the overall average according to the sample average in statistical inference.
This law is a special case of Chebyshev's law of large numbers, that is, when n is large enough, the frequency of event A will be almost close to its probability, that is, the stability of frequency.
In the sampling survey, the theoretical basis for estimating the total number by using the number of samples is here.
For ordinary people, the law of large numbers is not strictly stated as follows: X_ 1, ..., X_n is an independent and identically distributed sequence of random variables, with an average value of u, and S_n=X_ 1+...+X_n, then S_n/n converges to u. 。
If we say "weak law of large numbers", the above convergence is convergence with probability; if we say "strong law of large numbers", the above convergence is almost inevitable/convergence with probability 1.
Generally speaking, the law of large numbers means that when the number of samples is large, the sample mean and the true mean are completely close. Together with the central limit theorem, this conclusion has become one of the cornerstones of modern probability theory, statistics, theoretical science and social science, and its importance is not even weaker than calculus in my opinion. (Interestingly, although the expression and proof of the law of large numbers depends on modern mathematical knowledge, its conclusion first appeared before the appearance of calculus. And in life, even if there is no knowledge of calculus, it can be applied. For example, students who have never studied calculus can easily use excel or calculator to calculate statistics such as sample mean and apply them to social sciences. )
The earliest expression of the law of large numbers can be traced back to the Italian mathematician cardano around AD 1500. 17 13 years, the famous mathematician James (Jacob) Bernoulli formally proposed and proved the original law of large numbers. But at that time, the modern probability theory had not been established, and the tools of measure theory and real analysis had not appeared, so the law of large numbers at that time was based on the "probability of independent events". Later mathematicians such as Poisson (from which the name of the law of large numbers came), Chebyshev, Markov, Qin Qin (from which the name of the strong law of large numbers came), Borer and Canteli all contributed to the development of the law of large numbers. It was not until 1930 that Corgo Molov, the founder of modern probability theory and a master of mathematics, really proved the last powerful law of large numbers.
Suppose that x, X_ 1, ..., X_n are sequences of independent identically distributed random variables with an average value of u, and the law of large numbers for the sum of independent identically distributed random variables has the following form.
Elementary probability theory
(1). Weak law of large numbers with variance: If e (x 2) is less than infinity, then S_n/n-u convergence in probability reaches 0.
Proof method: Chebyshev inequality can be obtained. This proof was given by Chebyshev.
(2) Weak Law of Large Numbers with Mean: If u exists, then the convergence in probability of S_n/n-u reaches 0.
Proof method: Taylor expands the characteristic function, proves that it converges to a constant, converges according to the distribution, and then equates the convergence according to the distribution with the convergence in probability.
Modern probability theory
(3). Exact weak law of large numbers: If xP(| X | >；; X) when x tends to infinity and converges to 0, then S_n/n-u_n converges to 0 with probability, where u _ n = e [x 1 _ | x | 1 moment condition->; Torque-free condition; Strong law of large numbers: 4th moment condition->; Second moment condition->; 1 moment condition), the more difficult it is to prove.
Although only (3) and (6) are the most accurate results, it must be recognized that the development of mathematics is a gradual process. Under stronger conditions, the final law of large numbers cannot be obtained without the previous theorem. It took hundreds of years from the initial observation of the existence of the law of large numbers in nature to the final proof of the final form, and modern probability theory was also established in this process. In addition, although (3) and (6) are much stronger than the previous (1) and (5), the conditions of (1) and (5) are only the existence of second-order moments (or variances), so they have been widely used for hundreds of years, which is enough for general social science problems and statistical problems.
In a word, the law of large numbers contains the core knowledge in probability theory. Although the expression of "Four Certificates of the Law of Large Numbers" is vague and full of ridicule, it is not pedantic and worthless as implied by the "four-word writing of Hui" in Kong Yiji. As a graduate student majoring in probability or statistics, it is very beneficial to understand the differences between these theorems and their proof methods, as well as the work of predecessors' algebra. Of course, no one should memorize these proofs (I can't remember them myself), as long as I can understand them and understand the mystery.
Editing this passage is related to mathematician Laplace.
Laplace was born in Beaumont-Antoinette, Calva province, northwest France, on March 23rd, on 1749. He is a professor of mathematics at the Paris Military Academy, starting with 1795, and then a professor at the Paris Institute of Technology. 1799, he also served as the director of the French longitude bureau and served as the minister of the interior in Napoleon's government for six weeks. 18 16 was elected as an academician of the French Academy, and 18 17 was the president of the Academy. He died in Paris on March 5th, 827/KLOC-0.
In the process of studying celestial problems, Laplace created and developed many mathematical methods. Laplace transform, Laplace theorem and Laplace equation named after him are widely used in various fields of science and technology. [4]
Abraham de Morville
De Mofo, formerly known as Abraham de Moivre in French, (1667.05.26 France-1754.11.27 London, England), is a French mathematician. Demovo's most famous contributions to mathematics are Demovo formula and Demovo-Laplace central limit theorem, as well as his research on normal distribution and probability theory. Demover also wrote a textbook on probability, Opportunism, which is said to be highly praised by gamblers. Demover is one of the pioneers of analytic geometry and probability theory; He also found an approximate formula of binomial distribution for the first time, which is considered to be the first appearance of normal distribution.