e(X)= X 1 * p(X 1)+X2 * p(X2)+……+Xn * p(Xn)= X 1 * f 1(X 1)+X2 * F2(X2)+……+Xn * fn(Xn)

x； 1，X； 2，X； 3、……、X .

N is a discrete random variable, and p(X 1), p(X2), p(X3), ... p (xn) are the probability functions of these data. P(X 1), p(X2), p(X3), ... p (Xn) in several random data is understood as the frequency f(Xn), X2, X3, ..., xn of data X 1.

Application:

1, economic decision-making

Suppose a supermarket sells a certain commodity, the value range of weekly demand X is 10-30, and the value range of commodity purchase quantity is 10-30. The supermarket can make a profit for every commodity it sells, 500 yuan. If the supply exceeds the demand, the price will be reduced, and every product processed will lose money 10 yuan. 0 yuan; If the supply exceeds the demand, you can change hands from other supermarkets. At this point, supermarket goods can make a profit in 300 yuan. Can the supermarket get the maximum profit when calculating the purchase quantity? Expectation of maximum profit.

Analysis: Because the demand (sales volume) x of commodities is a random variable, which is evenly distributed in the interval, the sales profit value y of commodities is also a random variable. It is a function of x, which is called a random variable function. 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. Competition issues

Table tennis is our national game, and the military ball in the last century also brought some diplomacy to China. China has an absolute advantage in this sport. This paper raises a question about the arrangement of table tennis matches: suppose Germany (German player Bohr also has many fans in China) and China play table tennis. There are two kinds of competition systems, one is that each side has three winners, the other is that each side has five winners, and the other is that each side has five winners. Which is better for China?