Xiaodong Lee
By constructing an appropriate probability model, some commonly used inequalities are proved by using the properties, theorems and formulas of probability, which shows that the idea of probability method is efficient, concise and practical in solving problems. This paper expounds the application of probability method in inequality proof, shows the ingenuity and superiority of probability application, provides a new tool for solving some inequalities, and broadens the idea of inequality proof.
Keywords: probability method; Random variable; probability distribution
1 preface
Probability theory is a branch of mathematics, and it is very feasible and important to prove some inequalities by probability method. Academician Wang Zikun, a famous mathematician, pointed out in reference [3]: "It is one of the important research directions of probability theory to prove some inequalities or solve other problems in mathematical analysis by probability method." This paper enumerates several examples, constructs appropriate probability models according to their respective characteristics, and selects appropriate probability properties and theorems.
2 using the nature of probability, "any event has it." Prove inequality.
For a kind of inequality, it can be proved by the following probability idea: find all the independent variables in inequality and make them correspond to the probability of a random event respectively.
Example 1 Proof: If, then.
Prove that the two events are independent of each other, there are
[4]
because
therefore
that is
3 Prove inequality with ""
According to the definition of variance of random variables
Ueyuki
Example 2 is set to any real number, which proves that:
It is proved that the conclusion is transformed into a discrete random variable according to the definition of mathematical expectation, and its probability distribution is
rule
,
allow
that is
Generally speaking, when one side of the inequality is two series, where the square of the product is (where) and the other side is the sum or (where) of the square product of one series and another series, it can be proved by the following probability thought: construct a discrete random variable so that its probability distribution is
or
Then prove the inequality with mathematical expectation inequality.
4. Prove the inequality with "when the sum of random variables is independent of each other".
According to the nature of mathematical expectation and "if random variables are independent of each other, they exist", we can get
As a result, therefore
Example 3 demonstrates that:
It is proved that if the conclusion is transformed into that the sum of random variables is independent of each other, then the probability distribution of is
The probability distribution of is
rule
,,,
because
therefore
that is
.
5 Prove the inequality by Cauchy-Schwartz inequality [5].
Quadratic function considering real variables
Because everything has it, the quadratic equation has either no real root or only one multiple root, so we know that its discriminant is not positive, that is,
namely
Example 4 Assumption, Proof:.
It is proved that the probability distribution of the sum of random variables is
The probability distribution of is
The probability distribution of is
rule
,,
By, by
that is
.
6 Using monotonicity of probability to prove inequality
By knowing, and then by the finite additivity of probability, we get
We can also know from the nonnegativity of probability, so this is the monotonicity of probability: for events, if, then there is.
Example 5 Hypothesis, that is, proof:
Prove that hypothetical events,,, are independent of each other, and
,,,
because
From the monotonicity of probability
Because,,, are independent of each other, so
therefore
and
In the same way; In a similar way
Substitute the above three formulas into the formula to get the final product.
7 abstract
As can be seen from the above example, the key to prove some inequalities by probability method is to establish a probability model according to the concrete form of inequality, and then prove it by using the relevant properties and theorems of probability theory. On the one hand, it can provide a probability background for learning advanced mathematics and communicate the links between different disciplines and different aspects. On the other hand, the idea of solving problems is novel and unique, which is one aspect of the application of probability method in other branches of mathematics. In addition, the probability method can also be applied to.
refer to
[1] Wang zikun, the basis of probability theory and its application [M]. Beijing: Science Press, 1979.
[2] Yan, Liu Xiufang, Probability Theory and Mathematical Statistics [M]. Beijing: Higher Education Press, 1990.
[3] Sheng Shu, Xie Shiqian, Pan, Probability Theory and Mathematical Statistics [M]. Beijing: Higher Education Press, 200 1.
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