Probability theory and Mathematical Statistics study notes

Source: Internet
Author: User

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

Contact Us

The content source of this page is from Internet, which doesn't represent Alibaba Cloud's opinion; products and services mentioned on that page don't have any relationship with Alibaba Cloud. If the content of the page makes you feel confusing, please write us an email, we will handle the problem within 5 days after receiving your email.

If you find any instances of plagiarism from the community, please send an email to: info-contact@alibabacloud.com and provide relevant evidence. A staff member will contact you within 5 working days.

A Free Trial That Lets You Build Big!

Start building with 50+ products and up to 12 months usage for Elastic Compute Service

  • Sales Support

    1 on 1 presale consultation

  • After-Sales Support

    24/7 Technical Support 6 Free Tickets per Quarter Faster Response

  • Alibaba Cloud offers highly flexible support services tailored to meet your exact needs.