1, content difference: Chebyshev's law of large numbers describes that the mean of a series of independent variables (which can be distributed differently) converges to a constant, but it requires that the expectation and variance of each variable exist and are limited, and the additional condition that the mean of variance is infinitesimal in the higher order of the sample number n is satisfied. Hinchin's law of large numbers aims at the case that the mean of a series of independent identically distributed random variables converges to a constant, provided that the absolute expectation of distribution exists and is finite.
2. Variable relation: Chebyshev's law of large numbers pays more attention to the dependence of non-zero integers, indicating that any inequality can be transformed into such a formula. If the coefficients of the items on the right side of the equal sign are 1 in turn, a new inequality is obtained. Qin Xin's law of large numbers focuses more on the relationship between prime numbers and natural numbers, that is, every natural number not less than 1 can be written as the product of two prime numbers not less than itself.
3. Historical significance: Bernoulli's law of large numbers is the first strictly proved law of large numbers in human history, while Chebyshev's law of large numbers and Sinchin's law of large numbers are special cases of Bernoulli's law of large numbers. Because Bernoulli's law of large numbers has certain historical significance, and binomial distribution law of large numbers is the most common in daily life, textbook writers like to list this law of large numbers separately.
4. Scope of application: In daily life and practical application, binomial law of large numbers is the most common, while Chebyshev's law of large numbers has a relatively small scope of application.
The Relationship between Prime Numbers and Natural Numbers in Qin Xin's Law of Large Numbers;
Qin Xin's law of large numbers is a mathematical law about the relationship between prime numbers and natural numbers. This law points out that every natural number greater than 1 can be written as the product of two prime numbers not less than itself, which is also the source of the name of Qin Xin's law of large numbers.
This law has important application value in mathematics because it reveals the distribution law of prime numbers in natural numbers. Specifically, Qin Xin's law of large numbers shows that although prime numbers appear less frequently in natural numbers, they play a key role in describing the structure of natural numbers.
In order to better understand the relationship between prime numbers and natural numbers in Qin Xin's law of large numbers, we can consider a simple example. Suppose we have a natural number greater than 1, such as 100. We can decompose 100 into the product of two prime numbers, that is, 100= 13×7. In this example, 13 and 7 are two prime numbers, and the combination of their products just illustrates the content of Qin Xin's law of large numbers.
In fact, for any natural number n greater than 1, two prime numbers p and q can be found, so that n=pq. The proof of this property is one of the core contents of Qin Xin's law of large numbers. By proving this property, we can better understand the distribution law of prime numbers in natural numbers and further study the nature and structure of natural numbers.