Probability theory and Mathematical Statistics study notes

Chapter I. Stochastic events and probabilities

Chapter two stochastic variables and their distributions

Chapter three multivariate random variables and their distributions

The fourth chapter law of large numbers and the central limit theorem

The fifth chapter statistic and its distribution

The sixth chapter parameter estimation

The seventh chapter hypothesis test

Eighth chapter analysis of variance and regression analysis

Chapter I. Stochastic events and probabilities

1.1 Random events and their operations

The object of probability theory and mathematical statistic research is stochastic phenomenon. Probability theory is a model to study stochastic phenomena (i.e. probability distribution), and mathematical statistics is the data collection and processing of random phenomena.

Stochastic phenomenon: In certain conditions, the phenomenon that does not always appear the same result is called stochastic phenomenon.

Sample space: A collection of all possible basic results of a random phenomenon called a sample space

Random event: A set of random occurrences of certain sample points called random events

Random variables: Variables used to represent the results of random phenomena are called random variables

Relationship between events: inclusive, equal, incompatible

Operation of events: and, intersection, difference, opposition

Opposites, incompatible

The operational nature of events: commutative law, binding law, distributive law, duality law

Event fields

1.2 The definition of probability and its determination method

The axiomatic definition of probability:

1. The axiom of non-negativity

2. The axiom of regularization

3. Optional axiom of inclusion

Arranging and combining formulas

1. Multiplication principle

2. Principle of addition

Permutations and combinations

1. Arrange P (r,n)

2. Repeat arrangement N^r

3. Combination C (R,n)

4. Repeating combination C (R, N+r-1)

Frequency method for determining probability

Classical methods for determining probabilities

1.3 Nature of probability

The additive nature of probability

The monotonicity of probability

The additive formula of probability

Continuity of probabilities

1.4 Probability of a piece

Multiplication formula

Full probability formula

Bayesian formula

1.5 Independence

The occurrence of one event does not affect the occurrence of another event

Chapter two stochastic variables and their distributions

2.1 Random variables and their distributions

The distribution function of random variables

Monotonicity of

Boundedness

Right continuity

Probability distribution column of discrete random variables

Basic properties of distribution columns

1. Non-negative

2. Regularization

2.2 Mathematical expectation of random variables

The nature of mathematical expectation

2.3 Variance and standard deviation of random variables

1 Chebyshev Inequalities

2.4 Common discrete distributions

Two item distributions

Two-point distribution

Poisson distribution

Super Geometric distribution

Geometric distribution

2.5 Common continuous distribution

Normal

Evenly distributed

Exponential distribution

Gamma distribution

Beta distribution

2.6 Distribution of random variable functions

2.7 Other characteristics of the distribution

K-Order Moment

K-Order Original point moment

K-Order Center moment

Coefficient

Number of Bits

Number of Median

Coefficient of skewness

Coefficient of kurtosis

Chapter three multivariate random variables and their distributions

3.1 Multi-dimensional stochastic variables and their joint distributions

Multidimensional random variables

Union distribution function

Federated distribution Columns

Joint density function

Multi-item Distribution

Multidimensional hypergeometric distribution

Multidimensional Uniform distribution

Binary Normal distribution

3.2 Marginal distribution and independence of random variables

3.3 Distribution of multi-dimensional random variable functions

3.4 Characteristics of multi-dimensional random variables

3.5 article distribution and condition expectation

The fourth chapter law of large numbers and the central limit theorem

Two kinds of convergence of 4.1 random variable sequences

The fifth chapter statistic and its distribution

5.1 General and sample

The sixth chapter parameter estimation

Concept and unbiased nature of 6.1-point estimation

6.2 Moment Estimation and consistency

6.3 Maximum likelihood estimation and EM algorithm

6.4 Unbiased estimation of minimum variance

6.5 Bayesian estimates

6.6 Interval Estimation

The seventh chapter hypothesis test

Probability theory and Mathematical Statistics study notes