Self review ideas of probability theory

Review ideas of probability theory （ There are mistakes ）

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Basic concepts

1. sample space , And events 、 Poor event

2. The relationship between the two events ： Phase incompatibility 、 Is it opposite 、 The relationship between the two （$\rho$ The correlation coefficient only reflects the linear aspect , There may also be nonlinear relationships ）

3. Probability and relationship of event occurrence ： For example, the probability is 0 It does not necessarily represent an impossible event , The probability of impossible event must be 0. The same analogy to 1 And inevitable events

4. Conditional probability ：

   A priori probability and a posteriori probability ： One event occurs based on another event , Another event occurs based on the known occurrence of this event

   Bayes' formula + All probability formula P19

5. Event independence ： Judge whether it is independent , And whether the probability of occurrence of events is equal to the product of two events

Random distribution of variables

1. discrete ：$P\{X=x_k\}=p_k$

   Common discrete random variables ：0~1 Distribution , Bernoulli's test （ The binomial distribution ）、 Poisson distribution

   Distribution function ： Pay attention to some properties , Monotone increasing 、 Tend to be $+\infty$ by 1, Right continuous

2. Continuous type ：$F(x)={\int }_{-\infty }^{x}f(t)dt$

   Probability density 、 Distribution function

   Common continuous random variables ： Uniform distribution 、 An index distribution （ No memory ）、 Normal distribution P45

3. The distribution of functions of random variables ：

   Combine basic random variables with functions , For discrete form , Directly calculate the column distribution rate of possible values ; For continuous type , The function can be substituted into the distribution function of the basic random variable, and then the final probability density can be obtained by derivation (P53)

Multidimensional random variable distribution

discrete

1. Joint distribution rate ： Direct corresponding multiplication
2. Edge probability density ： It can be understood as the probability of occurrence of a certain value of one of the variables , That is, the joint distribution rate is added in one column or row

Continuity

1. Joint probability density ： When multiplying, pay attention to the definition field
2. Edge distribution ： Pay attention to the choice of integral region ,x、y In the area formed by the coordinate axis
3. The functional distribution of two random variables ：X*Y,X/Y Remember to add absolute value

Digital features

Mathematical expectation

expect ： A weight * probability

variance

$$D(X)=Var(X)=E\{[x-E(x){]}^2\}=E(X^2)-[E(X){]}^2$$

$$\begin{gathered} D(X+Y)=D(X)+D(Y) \\ D(X-Y)=D(X)+D(Y) \end{gathered}$$

Covariance coefficient

$\rho$ The correlation coefficient is only for the linear relationship , When they are independent of each other, they are aimed at the general relationship

Moment 、 Covariance matrix

The moment of origin ： First and second order origin moment

The central moment ： The second central moment is the variance

Law of large Numbers

Probability convergence is different from ordinary convergence , It is that the convergence can exceed the corresponding range at some time , The convergence is after a certain value , Has been small $ϵ$ Inside .

Chebyshev inequality ：

Central limit theorem ： Regularization idea of normal distribution

De Moivre-Laplace Theorem ： Only applicable to binomial distribution

Sampling distribution 、 It is estimated that 、 Hypothesis testing

$\chi$ Distribution 、$t$ Distribution 、$F$ Distribution

Common statistics ： Does not contain unknown parameters

Sampling transformation of several common statistics

On $\alpha$ quantile

The distribution of sample mean and sample variance of normal population

Parameter estimation

Moment estimate ：

Estimate the first-order moment of the population by using the first-order moment of the sample , For one parameter, only the first moment of the sample needs to be calculated , The second-order moment of the sample estimates the second-order moment of the population . For example, the title has two unknown parameters , Then I gave the sample , If you know the distribution , We can calculate the first and second moments of the sample by calculating the mean and variance of the sample , Then estimate the first and second moments of the population . Then it corresponds to the first and second moments of the two parameters of the population , Calculate the relationship between those two parameters and the first and second order of the sample

Maximum likelihood estimation ：

Calculate the probability density with unknown parameters , And then n Multiply the probability of samples , Is the likelihood function , The value of the parameter when the maximum value is obtained is obtained by derivation . Multivariable is to find partial differential .

Estimated evaluation criteria : unbiasedness 、 effectiveness 、 Consistency

interval estimation

Use those statistics of sampling distribution , For example, what kind of chi square distribution does the sample variance become 、t Distribution , Then according to the newly constructed sampling distribution , Use the given significance table to find the boundary , Pay attention to one side 、 Bilateral interval , Then simplify the interval , Get the confidence interval of the parameter

Known variance , Don't know the average ; Known mean ; I don't know the variance ……P175

Hypothesis testing

Similar to interval estimation , Also using sampling distribution statistics , Construct sampling distribution , Use significance test , Get the interval range , Then decide whether to accept or reject the hypothesis according to whether the assumed mean falls within or outside the confidence interval .

To be true ： Suppose it is true , Refuse

Take the false ： The assumption is false , Accept