The main contents of random events and probability check are:
(1) the relationship between events and operations, and use them for probability calculation;
Knowledge points and test sites of probability theory and mathematical statistics
Chapter 1 Knowledge Points: 18
1. 1 randomized trial: three characteristics of randomized trial.
(1) sample space: sample space; Sample point;
(2) Random events: random events; The event happened; Basic events; Inevitable events; An impossible event;
(3) Relationship between event and event operation: including relationship; Equal relationship; Incompatible with each other; Summer activities, product activities,
Poor events and opposing events;
(4) the operation law of the event.
1.2, the definition and operation of probability:
(1) frequency definition; (2) statistical definition of probability, (3) axiomatic definition of probability, (4) classical probability and (5) geometric probability.
1.3, conditional probability:
(1) definition; (2) nature; (3) multiplication formula. (4) Total probability formula, (5) Bayesian formula; ,
1.4 independence of events: (1) the nature that two events are independent of each other; (2) Independent definition of three (more) events, (3) Bernoulli test model.
Test center: 1, the representation and operation of events, 2, the proposition about the basic properties of probability, 3, the calculation of classical probability,
4. Calculation of geometric probability, 5. The proposition of event independence, 6. Calculation of conditional probability and product event probability,
7. Propositions of total probability formula and Bayesian formula, 8. Bernoulli test.
Chapter II Knowledge Points: 19
2. The definition of1(1) random variable; (2) The distribution function of random variables and its properties.
2.2 Discrete random variables and their probability distribution:
The definition of (1) discrete random variable;
(2) the distribution law of discrete random variables;
Several common discrete random variables: (1) (0- 1) distribution; (2) binomial distribution; (3) Poisson distribution;
(4) Hypergeometric distribution; (5) geometric distribution; (6) Pascal distribution,
Master the model of each distribution and write its distribution law or distribution density.
2.3 Continuous random variables and their probability distribution:
Definition of (1) distribution function;
(2) Basic properties of distribution function;
(3) The relationship between the distribution function and the distribution law of discrete random variables;
(4) Definition of probability density of continuous random variables;
(5) the nature of probability density;
Several common continuous random variables
(1) uniform distribution: (1) probability density; (2) distribution function;
(2) Orthotropic distribution: (1) probability density; (2) distribution function;
2.4 Function Distribution of Random Variables
Function Distribution of (1) Discrete Random Variables
(2) Function distribution of continuous random variables
Test site: 1, propositions about basic concepts such as distribution law, distribution function and distribution density,
2. Proposition about the relationship between distribution law, distribution density and distribution function,
3. Know the probability of the event, calculate the parameters in the event, 4. Calculate the probability of related events by using common distribution,
5. Find the distribution law, density and function of random variables; 6. Find the distribution of random variable function.
Chapter III Knowledge Points: 13
3. 1 Multidimensional Random Variables and Their Distribution
The definition of (1) two-dimensional random variable (1);
(2) The definition and basic properties of (1) joint distribution function of two-dimensional random variables; (2) Definition and basic properties of edge distribution function.
(3) Discrete two-dimensional random variables: (1) joint distribution law, (2) edge distribution law, (3) distribution function;
(4) Continuous two-dimensional random variables: (1) joint probability density, (2) marginal probability density and (3) related properties.
(5) Generalization: (1) n-dimensional random variables and their distribution.
3.2 Conditional distribution of two-dimensional random variables (no talking, no testing)
3.3 (1) Definition of independence of two-dimensional random variables;
3.4 Functions of two random variables and their distribution: (1) Probability distribution of functions of two discrete random variables,
(2) Probability distribution of two continuous random variable functions (mainly sum and maximum)
Test site: 1. A proposition on the basic concepts and properties of two-dimensional random variables and their distribution.
2. There are given experiments to determine various probability distributions.
3. Calculation of a new joint distribution of two-dimensional random variables defined by a given event or random variable,
4. Find the edge distribution from the given joint distribution or joint density,
5. Using the known distribution and independence to calculate the probability of related events, 6. Find the distribution of random variable function,
7. Independence of random variables.
Chapter IV Knowledge Points: 15
4. Definition of mathematical expectation of1(1) discrete random variables; (2) The definition of mathematical expectation of continuous random variables;
(3) Mathematical expectation of random variable function; (D) the nature of mathematical expectations
4.2 Definition of (1) variance of random variables; (2) standard deviation; (3) nature. (4) Variance of discrete and continuous random variables; (5) Calculation formula of variance;
4.3 Mathematical expectation and variance of (1Poisson distribution, (2) Mathematical expectation and variance of uniform distribution, (3) Mathematical expectation and variance of exponential distribution; (4) mathematical expectation and variance of binomial distribution, (5) mathematical expectation and variance of normal distribution;
4.4 (1) Definition and calculation of covariance and correlation coefficient; (2) Definition and calculation of torque.
Test site: 1, find the expectation and variance of discrete random variables, 2, find the expectation and variance of continuous random variables,
3. Find the expectation and variance of random variable function; 4. Discuss and calculate covariance, correlation coefficient and moment.
Chapter 5 Knowledge Points: 5
5. 1 law of large numbers
(A) Chebyshev inequality and its application
(2) (1) Bernoulli's law of large numbers, (2) Chebyshev's law of large numbers
5.2 Central Limit Theorem
The central limit theorem of (1) independent identical distribution;
(2) Demovo-Laplace theorem and its application examples.
Test site: 1, propositions about Chebyshev inequality and law of large numbers, 2, propositions about central limit theorem.
Chapter VI Knowledge Points: 10
6. 1 random sample: (1) population, individual, simple random sample, sample value, etc. (2) the definition of statistics;
Several commonly used statistics: (1) sample mean, (2) sample variance and (3) sample standard deviation. (4) the origin moment of the sequential sample, and (5) the center moment of the sequential sample.
6.2 Sampling distribution: (1) distribution, (2) distribution (student distribution) and (3) distribution of common statistics.
Test site: 1. Find the joint distribution function of samples, 2. Find the numerical characteristics of statistics, 3. Find statistical distribution.
4. Find the probability of statistical values and the capacity of samples.
Chapter 7 Knowledge Points: 12
7. 1 parameter point estimation method: (1) moment estimation method; (2) Maximum likelihood estimation method
Likelihood function: discrete; Continuous type;
7.2 Evaluation criteria for point estimation
(1) (1) Unbiasedness, (2) Validity and (3) Consistency (self-study)
7.3 interval estimation
The concept of (1) interval estimation: (1) confidence interval and confidence level; Number of pivots.
(2) (1) The step of finding the confidence interval of the unknown parameter.
(3) Interval estimation of mean and variance of normal population (only a single normal population)
Confidence interval of (1) mean; (2) Confidence interval of variance; (3) unilateral confidence interval;
Test site: 1, moment estimation and maximum likelihood estimation, 2, discussion on the selection criteria of estimator,
3. Find the interval estimation of parameters.
Chapter 8 Knowledge Points: 10
8. 1 (1) Basic concept of hypothesis testing: (1) test statistics; Original hypothesis; Substitution hypothesis; Deny domain; (2) Two types of errors;
(2) (1) the procedure of hypothesis testing;
8.2 (1) Hypothetical Test of Single Normal Population Mean
(1) known, test (z test) (2) unknown, test (t test)
(3) Hypothesis test of single normal population variance.
(1) unknown, test (test) (2) known, test (test)
Two types of hypothesis testing should be distinguished: (1) two-sided hypothesis testing, (2) left hypothesis testing and (3) right hypothesis testing.
Test location: 1, hypothesis test of single normal population mean,
2. Hypothesis test of single normal population variance.
(2) the definition and nature of probability, which is used to calculate the probability of some events;
(3) Classical probability and geometric probability;
(4) calculating the probability by using addition formula, conditional probability formula, multiplication formula, total probability formula and Bayesian formula;
(5) The concept of event independence is used to calculate the probability of an event;
(6) Calculation of independent repeated test, Bernoulli probability and related event probability.
Candidates are required to understand the basic concepts, analyze the event structure, use formulas correctly, master some skills and calculate probability skillfully.
The main contents of random variables and probability distribution are:
(1) Use the definition and properties of distribution function, probability distribution or probability density for calculation;
(2) Grasp the distribution and properties of some important random variables, mainly: (0- 1) distribution, binomial distribution, Poisson distribution, geometric distribution, hypergeometric distribution, uniform distribution, exponential distribution and normal distribution, and calculate the probability of related events;
(3) Find the function distribution of random variables.
(4) Find the distribution of simple functions of two random variables, especially the distribution of the sum of two independent random variables.
Candidates are required to master the calculation of distribution function, edge distribution and conditional distribution, master the method of judging independence and carry out relevant calculations, and will find the distribution of two random variable functions.
The main contents of digital characteristics of random variables are:
(1) Definition, properties and calculation of mathematical expectation and variance;
(2) Mathematical expectation and variance of common random variables;
(3) Calculate the mathematical expectation and variance of some random variable functions;
(4) Definition, properties and calculation of covariance, correlation coefficient and moment;
Candidates are required to master the definition, nature and calculation of mathematical expectation and variance, the method of determining the distribution of random variables through given experiments, and then calculate the characteristics of correlation number, covariance, correlation coefficient and moment, and master the method of judging that two random variables are irrelevant.
The main contents of the law of large numbers and the central limit theorem are as follows:
(1) Chebyshev inequality;
(2) Law of large numbers;
(3) Central limit theorem.
Candidates are required to prove the inequality with Chebyshev inequality and approximate the probability of events with central limit theorem.
The main contents of the basic concepts of mathematical statistics are:
(1) The concepts, properties and calculation of sample mean, sample variance and sample moment;
(2) The definitions, properties and quantiles of χ 2 distribution, T distribution and F distribution;
(3) Derive the distribution of some statistics (especially some statistics of normal population) and calculate the correlation probability.
Candidates are required to master the properties and calculation of sample mean and sample variance, and the distribution of some statistics about the normal population will be deduced according to the definitions and properties of χ2 distribution, T distribution and F distribution.
The main contents of parameter estimation check are:
(1) Find the moment estimation and maximum likelihood estimation of parameters;
(2) To judge the unbiased, effective and consistent of the estimator;
(3) Find the confidence interval of normal population parameters.
Candidates are required to obtain the moment estimation, maximum likelihood estimation and unbiased judgment of parameters skillfully, and the confidence interval of normal population parameters will be obtained.
The main contents of hypothesis testing are:
Significance test of (1) normal population parameters;
(2) χ2 test of population distribution hypothesis.
Candidates are required to test the significance of normal population parameters and χ2 test of population distribution hypothesis.
Common types of questions are: fill-in-the-blank questions, multiple-choice questions, calculation questions and proof questions. The main types of problems are:
(1) Determine the relationship between events and perform the operation of events;
(2) probability calculation by using the relationship of events;
(3) Using the properties of probability to prove the probability equation or calculate the probability;
(4) Probability calculation of classical probability and geometric probability;
(5) Calculating probability by using addition formula, conditional probability formula, multiplication formula, total probability formula and Bayesian formula;
(6) The proof and calculation probability of event independence;
(7) Single repeated test and calculation of Bernoulli probability formula;
(8) Using the definition and properties of distribution function, probability distribution and probability density of random variables, determine unknown constants or calculate probabilities;
(9) Find out the distribution of random variables from the given experiment;
(10) Use common probability distributions (such as (0- 1) distribution, binomial distribution, Poisson distribution, geometric distribution, uniform distribution, exponential distribution, normal distribution, etc.) to calculate the probability. );
(1 1) Find the distribution of random variable functions.
(12) Determine the distribution of two-dimensional random variables;
(13) Calculate the probability by using two-dimensional uniform distribution and normal distribution;
(14) Find the edge distribution and conditional distribution of two-dimensional random variables;
(15) to judge the independence and calculation probability of random variables;
(16) Find the distribution of two independent random variable functions;
(17) Use the definition, nature and formula of the mathematical expectation and variance of random variables, or use the mathematical expectation and variance of common random variables to find the mathematical expectation and variance of random variables;
(18) Find the mathematical expectation of random variable function;
(19) Find the covariance and correlation coefficient of two random variables and judge the correlation;
(20) Find the moment and covariance matrix of random variables;
(2 1) Deriving Probability Inequality by Chebyshev Inequality;
(22) Approximate calculation of probability by using the central limit theorem;
(23) Using the definitions and properties of T distribution, χ2 distribution and F distribution, the distribution and properties of statistics are deduced;
(24) Inferring the distribution of some statistics (especially normal population statistics);
(25) calculate statistical probability;
(26) Find the moment estimator and maximum likelihood estimator of unknown parameters in the population distribution;
(27) Judge the unbiased, effective and consistent estimators;
(28) Find the confidence interval of one or two normal population parameters;
(29) test the significance of one or two normal population parameter assumptions;
(30) Use χ2 test to test the hypothesis of population distribution.
This part mainly investigates the basic concepts, properties and theories of probability theory and mathematical statistics, and investigates the application of basic methods. Through the analysis of the test questions over the years, we can see that the test questions of probability theory and mathematical statistics, even fill-in-the-blank questions and multiple-choice questions, only test a single knowledge point, and most of the test questions are aimed at testing candidates' understanding ability and comprehensive application ability. Candidates are required to use their knowledge flexibly, establish a correct probability model, and comprehensively use knowledge such as limit, continuous function, derivative, extreme value, integral, generalized integral and series to solve problems.
When answering this part of the exam questions, the mistakes that candidates are prone to make are:
The concept of (1) is unclear, and the relationship between events and the structure of events are unclear;
(2) The analysis of the test is wrong, and the probability model is wrong;
(3) Improper use of probability calculation formula;
(4) Unable to skillfully use independence proof and calculation;
(5) Unable to master and use commonly used probability distribution and its numerical characteristics;
(6) The relevant definitions, formulas and properties cannot be correctly applied for comprehensive analysis, operation and proof.
According to the answers of candidates over the years, the scoring rate of probability theory and mathematical statistics is about 0.3, and the discrimination is generally above 0.40. This shows that the test questions are both difficult and highly differentiated.