When the discrete random variable X takes a countable value, its mathematical expectation requires the series ∑| Xi |π to converge, otherwise the mathematical expectation does not exist; If a continuous random variable takes a value in an infinite interval, its mathematical expectation is generalized integral, which requires absolute convergence, otherwise the mathematical expectation does not exist. For example, the mathematical expectation ex of Cauchy distribution does not exist.

Mathematical expectation (or mean, or expectation for short) is the sum of the possible results multiplied by the results in each experiment, and it is one of the most basic mathematical characteristics. It reflects the average value of random variables.

It should be noted that the expected value is not necessarily equal to the common sense "expectation"-"expected value" is not necessarily equal to every result. The expected value is the average of the output values of variables. The expected value is not necessarily contained in the set of output values of variables.

The law of large numbers stipulates that as the number of repetitions approaches infinity, the arithmetic average of numerical values almost inevitably converges to the expected value.

Extended data:

Application of Mathematical Expectation

1, economic decision-making

Suppose that the weekly demand of a commodity sold in a supermarket is in the range of 10 to 30, and the purchase quantity of the commodity is in the range of 10 to 30 (only once a week). Everything sold in the supermarket can make a profit, 500 yuan. If the supply exceeds the demand, the price will be reduced, and the loss per unit of goods will be 100 yuan.

If the demand exceeds the supply, it can be transferred from other supermarkets. At this time, the goods in the supermarket can make a profit in 300 yuan. When trying to calculate the purchase quantity, the supermarket can get the best profit. And seek the expectation of maximum profit.

Analysis: Because the demand (sales volume) X of this commodity is a random variable, which is evenly distributed in the interval, and the profit value Y of selling this commodity is also a random variable, which is a function of X and is called a function of random variables. The best profit involved in the problem can only be the mathematical expectation of profit (that is, the maximum of average profit).

Therefore, the process of solving this problem is to determine the functional relationship between Y and X, then find the expected E(Y) of Y, and finally find the maximum point and maximum value of E(Y) by extreme value method.

2. Sports competition issues

Table tennis is our national game. In the last century, ice hockey also brought some diplomacy to China. China has an absolute advantage in this sport.

Now I want to ask a question about the arrangement of table tennis: Suppose the German national team (German star Bohr also has many fans in China) plays against China. There are two competition systems, one is three players per team, and the other is five players per team. Which is more beneficial to China?

Analysis: Because of the advantages of China team in this competition, let's assume that the winning percentage of each German player in China team is 60%, and then we only need to compare the corresponding mathematical expectations of the two teams.

Baidu Encyclopedia-Mathematical Expectation