In mathematics and statistics, the law of large numbers (also known as the law of large numbers) is a law that describes the results of repeated experiments. According to this law, the more samples, the higher the probability that the arithmetic mean is close to the expected value. The law of large numbers is very important because it "explains" the long-term stability of the mean of some random events.

We know that the law of large numbers studies the statistical regularity of random phenomena. When we repeat the same experiment a lot, the final experimental result may be stable near a certain value. Just like tossing a coin, when we keep throwing it thousands or even tens of thousands of times, we will find that the number of heads or tails will be close to half.

Significance of strong law of large numbers

This series of problems is actually the problem to be studied by the law of large numbers. At a very early time, people actually discovered this regular phenomenon, and many mathematicians have studied this phenomenon, including Bernoulli (later, in order to commemorate him, people thought that he was the first person to study this problem, in fact, mathematicians had studied it before him).

Bernoulli put forward a limit theorem in 17 13. At that time, this theorem had no name, and later it was called Bernoulli's law of large numbers. Therefore, the first limit theorem on the law of large numbers in the history of probability theory belongs to Bernoulli, which is the basic law of probability theory and mathematical statistics and belongs to the category of weak law of large numbers.

When an experiment is repeated a lot, the final frequency is infinitely close to the event probability. Bernoulli successfully expressed this phenomenon in real life through mathematical language and gave it exact mathematical significance. He made people have a new understanding and deeper understanding of this kind of problem, and pointed out the direction and played a leading role for later generations to study the law of large numbers.

It laid the foundation for the development of the law of large numbers. Besides Bernoulli, there are many mathematicians who have made important contributions to the development of the law of large numbers, and some even spent their whole lives, such as democritus, Lyapunov, Lindbergh, Ferrer, Chebyshev, Xin Qin and so on. These people have played an inestimable role in the progress of the law of large numbers and even probability theory.