In probability theory and mathematical statistics, mathematical expectation (or mean for short, or expectation) is the sum of the probabilities of every possible result multiplied by its result in the 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.

Economic decision:

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 find out the expected value of the 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.

Baidu Encyclopedia-Mathematical Expectation