Probability and Mathematical Statistics

I. General requirements

Understand the basic ideas of probability theory and mathematical statistics, understand the key concepts and ideas in the process of transforming classical probability into axiomatic probability, understand the statistical principle of estimation and test of mathematical statistics, master the probability calculation method and its application of classical probability model, and master the basic estimation and test methods.

Second, the content

1. Definition and operation of random events, definition and properties of probability.

1) Understand the concept of sample space (basic event space), understand the concept of random events, and master the relationship and operation of events;

2) Understand the concepts of probability and conditional probability, master the basic properties of probability, and calculate classical probability and geometric probability;

3) Master the addition formula, subtraction formula, multiplication formula, total probability formula and Bayesian formula of probability;

4) Understand the concept of event independence and master the probability calculation with event independence;

5) Understand the concept of independent repeated test and master the calculation method of related event probability.

2. One-dimensional random variables and their distribution

1) Understand the concept of random variables and the concept and properties of distribution functions.

2) Calculate the probability of events related to random variables.

3) Understand the concept and probability distribution of discrete random variables;

4) Master 0- 1 distribution, binomial distribution, geometric distribution, hypergeometric distribution, Poisson distribution and their applications;

5) Understand the concepts of continuous random variables and their probability density, and master uniform distribution, normal distribution, exponential distribution and their applications.

3. Multidimensional random vectors and their distribution

1) Understand the concept of multidimensional random variables and the concept and properties of the distribution of multidimensional random variables;

2) Understand the probability distribution, edge distribution and conditional distribution of two-dimensional discrete random variables;

3) Understand the probability density, edge density and conditional density of two-dimensional continuous random variables;

4) Probability of finding related events of two-dimensional random variables;

5) Understand the concepts of independence and irrelevance of random variables, and master the conditions of mutual independence of random variables;

6) Grasp the two-dimensional uniform distribution, understand the probability density of the two-dimensional normal distribution, and understand the probability meaning of the parameters;

7) The distribution of simple functions of two random variables can be found, and the distribution of simple functions of multiple independent random variables can also be found.

4. Digital characteristics of random variables

1) Understand the concept of digital characteristics of random variables (mathematical expectation, variance, standard deviation, moment, covariance, correlation coefficient);

2) Be able to use the basic properties of digital features and master the commonly distributed digital features.

5. Law of Large Numbers and Central Limit Theorem

1) Understand Chebyshev inequality, Chebyshev's law of large numbers, Bernoulli's law of large numbers and Sinchin's law of large numbers (the law of large numbers for independent and identically distributed random variable sequences);

2) Understand de moivre-Laplace Theorem (binomial distribution takes normal distribution as the limit distribution) and Levi-Lindbergh Theorem (central limit theorem of independent identically distributed random variable sequence).

6. Basic concepts of mathematical statistics

1) Understand the concepts of population, simple random sample, statistics, sample mean, sample variance and sample moment;

2) Understand the concepts and properties of distribution, t distribution and f distribution, and master the common sampling distribution theorem of normal population.

7. Parameter estimation

1) Understand the concepts of point estimation, estimator and parameter estimation.

2) Master the moment estimation method and maximum likelihood estimation method, understand the concepts of unbiased estimator, effectiveness (minimum variance) and consistency (consistency), and verify the unbiased estimator;

3) Understand the concept of interval estimation. We will find the confidence interval of the mean and variance of a single normal population, and the confidence interval of the mean difference and variance ratio of two normal populations.

8. Hypothesis test

1) Understand the basic idea of significance test and master the basic steps of hypothesis test.

2) Understand the two possible errors in hypothesis testing, and master the hypothesis testing of the mean and variance of single and two normal populations.

Third, the question type and score ratio

Multiple choice questions: (10%)

Fill in the blanks: (10%)

Short answer: (20%)

Calculation problem: (60%)