Mathematical expectation is one of the important numerical characteristics of random variables, and it is also one of the most basic characteristics of random variables. Through several examples, this paper expounds the application of teaching expectation in probability theory and mathematical statistics in life. This paper lists some real-life examples and expounds the important application of mathematical expectation in economic and practical problems.
Keywords: random variable, mathematical expectation, probability, statistics
Mathematical expectation, referred to as mean for short, is an important numerical feature in probability theory and has important applications in economic management. This paper discusses some simple applications of mathematical expectation in economic and practical problems, aiming to help students understand the rich background of the close relationship between knowledge and human practice and experience it for themselves. Is math really useful? .
1. Decision scheme problem
Decision-making scheme refers to the scheme with the greatest mathematical expectation as the best scheme for decision-making. It helps people make choices and decisions from possible schemes in complex situations. Specifically, if you know any scheme AI (I = 1, 2,? M) in each influencing factor SJ (j = 1, 2,? When it happens, we can compare the expected profits of various schemes and choose the scheme with the highest expected profit as the best scheme.
1. 1 investment plan
Suppose someone invests 654.38 million yuan a year, and there are two investment schemes: one is to buy stocks; Second, there are banks that earn interest. The income from buying stocks depends on the economic situation. If the economic situation is good, it can make a profit of 40 thousand, if the situation is moderate, it can make a profit of 1 10 thousand, and if the situation is bad, it will lose 20 thousand. If you deposit it in the bank, assuming the interest rate is 8%, you can get 8,000 yuan of interest. Assuming that the probability of good, medium and poor economic situation is 30%, 50% and 20% respectively. Which scheme should be chosen to make the investment more effective?
Application of Mathematical Expectation [Abstract] Discrete random variable mathematical expectation is one of the important concepts in probability theory and mathematical statistics, which is used to reflect the characteristic number of random variable value distribution. By discussing some simple applications of mathematical expectation in economic and practical problems, it is hoped that students can understand that the theoretical knowledge of mathematical expectation is closely related to human practice, and the two are inseparable and closely related.
[Keywords:] Mathematical expectation; Discrete random variable
First, the connotation of mathematical expectation of discrete random variables
In probability theory and statistics, the sum of the products of all possible values xi of discrete random variables and the corresponding probability P(=xi) is called mathematical expectation (let series converge absolutely), and it is recorded as E(x). Mathematical expectation, also known as expectation or mean, is actually the average of random variables and one of the most basic mathematical characteristics of random variables. But what is the strict definition of expectation? Xi * pi is absolutely convergent, and attention is absolute, that is, it is different from the commonly understood average. A random variable can have an average value or a median value, but its expected value does not necessarily exist.
Second, the role of mathematical expectations on discrete random variables
It is expected to represent the average value of random variables in random experiments, which is the average value in the sense of probability, which is different from the arithmetic average value of corresponding values. It is a generalization of simple arithmetic average, similar to weighted average. When solving practical problems, as an important parameter, it plays an important guiding role in the fields of market forecasting, economic statistics, risk and decision-making, sports competitions, etc., which has a far-reaching impact on the future study of advanced mathematics, mathematical analysis and related disciplines and lays a good foundation. As a statistical numerical feature in the basic theory of mathematics, it is widely used in engineering technology, economy and social fields. Its significance lies in solving the analysis method of mathematical model abstracted from practice, so as to achieve the purpose of understanding the laws of the objective world and provide accurate theoretical basis for further decision analysis.
Third, the solution of mathematical expectation of discrete random variables
Solving the mathematical expectation of discrete random variables is usually divided into four steps:
1. Determine the possible values of discrete random variables;
2. Calculate the probability corresponding to each possible value of the discrete random variable;
3. Prepare the distribution list and check whether the distribution list is correct;
4. Find expectations.
Fourth, the application of mathematical expectation
(A) the application of mathematical expectations in the economy
Example 1: Suppose Xiao Liu invests 200,000 yuan, and there are two investment schemes. Option 1: for investment in buying a house; Option 2: deposit in the bank to earn interest. The income from buying a house depends on the economic situation. If the economic situation is good, it can make a profit of 40 thousand, if the situation is moderate, it can make a profit of 1 10 thousand, and if the situation is bad, it will lose 20 thousand. If it is deposited in the bank, the interest rate is assumed to be 5. 1%, and the interest can be 1 1000 yuan. It is assumed that the probability of good, medium and poor economic situation is 40%, 40% and 20% respectively. Which scheme should be chosen to make the investment more effective?
The first investment plan:
The profit expectation of buying a house is: E(X)=4? 0.4+ 1? 0.4+( - 2)? 0.2= 1.6 (ten thousand yuan)
The second investment plan:
The bank's profit expectation is E(X)= 1. 1 (ten thousand yuan),
Because: e (x) > E(X)，
From the above two investment schemes, it can be concluded that the expected income from buying a house is greater than the expected income from the deposit bank, and the house buying scheme should be adopted. There are two investment schemes here, but the economic situation is an uncertain factor, and the choice is based on the level of mathematical expectations.
(B) the application of mathematical expectations in the company's needs
Example 2: A small company expects that the market demand will increase. The employees of this company are working at full capacity at present. In order to meet the market demand and increase output, the company considers two schemes: the first scheme: let employees work overtime; The second option: add equipment.
Suppose the company predicts that the probability of market demand increase is p, and of course the probability of market demand decrease is1-p. If the relevant data is known, it is listed in the following table:
Market demand minus (1-p) market demand increase (p)
Maintain the status quo (x)
200 thousand 240 thousand
Employee overtime (x)
19032000
Yao Jia Equipment (X)
150,000 340,000
According to the conditions, it is cost-effective to let employees work overtime or increase equipment when the market demand increases. But the reality is that we don't know what will happen, so we should compare the expected profits of several schemes. Judging from expectations:
E(X)=20( 1-p)+24p，E(X)= 19( 1-p)+32p，E(X)= 15( 1-p)+34p
There are two situations to check:
(1) when p=0.8, then E(X)=23.2 (ten thousand), E(X)=29.4 (ten thousand), E(X)=30.2 (ten thousand), so the company can decide to update equipment and expand production;
(2) When p = 0.5, E(X)=22 (ten thousand), E(X)=25.5 (ten thousand), E(X)=24.5 (ten thousand), at this time, the company can decide to take emergency measures for employees to work overtime to expand production.
Therefore, from the above two situations, it can be concluded that if p=0.8, the company can decide to update equipment and expand production. If p = 0.5, the company can decide to take emergency measures for employees to work overtime. Therefore, as long as the possibility of market demand growth is above 50%, the company should take certain measures to increase profits.
(C) the application of mathematical expectations in sports competitions
Table tennis is our national sport, and people all over the country especially like it. We have an absolute advantage in this sport. There are two schemes for the arrangement of table tennis competition system:
The first scheme is that each side has three players and wins two out of three games. The second scheme is that each side has five people and wins two out of five games. Which of the two schemes is more beneficial to China? Let's look at an example:
Let's say that the winning percentage of each member of the China team is 55% against each member of the United States team. According to the previous analysis, we only need to compare the mathematical expectations of the two teams.
In the best-of-five system, if China wants to win, there are three results: 3, 4 and 5. We use binomial law and probability theory to calculate the probability corresponding to three results, and just get the probability corresponding to three fields: 0.33465; The probability of just getting four corresponding fields: 0.2512; Probability of winning five games: 0.07576.
Let the random variable X be the number of games won by the China team under this competition system, then the distribution law of X can be established: X 3 4 5.
P 0.33465 0.25 12 0.07576
Calculate the mathematical expectation of random variable x;
E(X)=3？ 0.33465+4? 0.25 12+5? 0.07576=2.0465 1
In the best-of-three system, China won, and there were two or three results in the number of games won. The corresponding probability = 0.412; Probability of winning all three games =0.206.
Let the random variable Y be the number of games won by the China team under this competition system, then the distribution law of Y can be established:
X 2 3
Y 0.4 12 0.206
Calculate the mathematical expectation of random variable y;
E(Y)=2？ 0.4 12+3? 0.206= 1.2
Compare two expected values, e (x) >; Therefore, we can draw a conclusion that the best of five games system is more beneficial to China.
Therefore, in this kind of competition, the best of five games system is more beneficial to China. In sports competitions, we should look at the specific details and situations, grasp the competition system, use the knowledge we have learned, give full play to our expectations, and know ourselves and ourselves.
(D) Mathematical expectations of enterprise profit evaluation
In market economy activities, manufacturers' production or businessmen's sales always pursue profit maximization. In the production process, supply exceeds demand or supply exceeds demand, which is not conducive to maximizing profits and expanding reproduction. But in the market economy, it is always changing rapidly, and supply and demand are often uncertain. In general, manufacturers or merchants often make the best production activities or sales strategies based on past data, combined with the current specific situation and specific objects, by combining the mathematical expectation method with the relevant knowledge of calculus.
Suppose a company plans to develop a new product market and try to determine its output. It is estimated that if a product is sold, the company can make a profit of A yuan, while if a product is overstocked, it can lead to a loss of B yuan. In addition, the company predicts that the sales volume of product X is a random variable with a distribution of P(x). So, how to make the output of products, in order to get the maximum profit?
Suppose the company produces X pieces of this product every year, although X is certain. However, because demand (sales volume) is a random variable, income y is a random variable, which is a function of x:
When xy, y=Ax;
When xy, y = ay-b (x-y).
So the expected income becomes a problem:
When x is what value, the expected return can reach the maximum. Using the knowledge of calculus, it is not difficult to get.
The solution of this problem is to find the maximum and minimum values expected by the objective function.
(E) Mathematical expectations in insurance.
The probability that a family's valuables above 50,000 yuan are stolen in one year is 0.005. If the insurance company provides home insurance of 50,000 yuan or more a year, participants need to pay the insurance premium 200 yuan. If the property of more than 50,000 yuan is stolen within one year, the insurance company will pay one yuan (a & gt200). How to determine A can make the insurance company expect to make a profit?
Let x represent the income of the insurance company to any insured family, then the value of x is 200 or 200? A, its distribution list is:
X 200 200-a
p 0.995 0.005
E(x)=200？ 0.9958+(200-a)？ 0.005 = 200-0.005 a & gt； 0，a
From the above daily life, we can easily find that using the knowledge of mathematical expectation of discrete random variables is of great help to solve some practical problems in life.
Therefore, in real life, we should use the knowledge of mathematical expectation of discrete random variables to face the requirements of today's information age. We should be active in thinking and dare to innovate. We should not only learn the knowledge of mathematical reasoning, but also pay attention to the practical application of what we have learned, so as to combine reasoning with practice and apply what we have learned. Of course, it is only a part of the application of mathematics expectation in real life, and there are more applications waiting for us to think, discover and explore, and create more valuable things and wealth for our great era.