概率论

Probability and Statistics

1.Overview and Descriptive Statistics

population

a well-defined collection of objects

census

desired information is available for all objects in the population

a sample

a subset of the population

a variable

any characteristic whose value may change from one object to another in the population

univariate data

consists of observations on a single variable

bivariate data

consists of observations on two variables

multivariate data

consists of observations on more than variables

descriptive statistics

graphical

histograms

boxplots

scatter plots

numerical

means

standard deviations

correlation coefficients

inferential statistics

Techniques for generalizing from a sample to a populationThe population as consisting of all possible data that might be made under similar experimental conditions. Such a populationis referred to as a conceptual or hypothetical population. There are a number of problem situations in which we fit questions into the framework of inferential statistics by conceptualizing a population. 

The relationship between probability and inferential statistics

Collection of Data

Simple random sample

Stratified random sample

Visulization of Data

Stem and Leaf plot

dot plot

histogram

Measure of Location

样本均值

中位数

Measures of Variability

Range

difference between the largest and smallest sample values

Deviations from the mean

Sum of deviations

样本方差和样本平方差

Population variance

2.Probability

2.1 Sample space and event

An experiment

is any action or process that generates observations.

The sample spaces

is the set of all the possible outcomes of an experiment.

An event E

is a subset of S

A is said to occur if the resulting experimental outcome is contained in A.

An event is any collection (subset) of outcomes contained in the sample space. An event is simple if it consists of exactly one outcome and compound if it consists of more than one outcome

Certain event E

Impossible event

Venn diagrams

Operation laws

2.2 Axioms of Probability

A mapping P satisfying above axioms is called a probability function.

Properties

Proofs

2.3 Counting techniques

排列

组合

性质

2.4 Conditional Probability

全概率公式、贝叶斯公式

2.5 Independent Event

Two events A and B are independent if P(A|B)=P(A) and are dependent otherwise.

Proof

Independence of More Than Two Events

3.Discrete Random Variables and Probability Distributions

3.1 Random Variables

3.2 Probability Distributions for Discrete Random Variables

Properties of c.d.f F(x)

3.3 Expected Values

Rules of Expected Value

A Shortcut Formula

Rules of Variance

3.4 The Binomial Probability Distribution

Any random variable whose only possible values are 0 and 1 is called aBernoulli random variable.

x—发生次数；n—实验次数；p—发生概率

3.5 The Poisson Probability Distribution

u=np=实验次数*发生概率

3.6 Hypergeometric and Negative Binomial Distributions

Hypergeometric Distribution

x—取中数；n—抽取总数；M—所要的最大数；N—样本总数