Key review route of probability theory and mathematical statistics examination

The key review route of probability theory and mathematical statistics exam

Preface

Hope to pass a simple route , Achieve accurate and efficient preparation for tomorrow's exam . Don't talk much , To rush ！

The content is divided into two parts: probability theory and mathematical statistics , The middle series is the law of large numbers and central limit theorem in Chapter 5 .

MindMap

Probability theory part

Mathematical statistics

probability theory

Basic concepts

The content of this part , My suggestion is to look directly at my previous blog, Or reading books and other online classes ppt And so on. .

I won't enumerate the distribution function of random variables , You can deduce it directly through the distribution law or probability density

discrete

0-1 Distribution

X~b§

Distribution law

$$P\{X=k\}=p^k(1-p{)}^{1-k},k=1,0$$

X	0	1
p_k	1-p	p

Mathematical expectation

$$E(X)=p$$

variance

$$D(X)=(1-p)\cdot p$$

The binomial distribution

X~b(n, p)

Distribution law

$$P\{X=k\}=p^k(1-p{)}^{1-k}$$

Mathematical expectation

$$E(X)=np$$

variance

$$D(X)=n(1-p)\cdot p$$

Poisson distribution

X~π(λ)

Distribution law

$$P\{X=k\}=\frac{{\lambda }^k{e}^{-\lambda }}{k!},k=0,1,2...$$

Poisson's theorem

Is to use Poisson to approximate binomial ,np=λ
$$\underset{n\to \infty }{\lim }{C}_n^k(1-p_n{)}^{n-k}=\frac{{\lambda }^k{e}^{-\lambda }}{k!}$$

Mathematical expectation

$$E(X)=\lambda$$

variance

$$D(X)=\lambda$$

Continuous type

Uniform distribution

X~U(a, b)

Probability density

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expect

$$E(X)=\frac{a+b}{2}$$

variance

$$D(X)=\frac{(b-a{)}^2}{12}$$

An index distribution

X~E(θ)

Probability density

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expect

$$E(X)=\theta$$

variance

$$D(X)={\theta }^2$$

Normal distribution

X~N(μ, σ)

Probability density

$$f(x)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{(x-u{)}^2}{2{\sigma }^2}},-\infty <x<\infty$$

Standard normal distribution

$$\begin{gathered} X\sim N(0,1^2) \\ \phi (x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2} \end{gathered}$$

Expectation and variance , In general, as long as it is converted into a standard normal distribution , Then it can be solved with the variance and expectation of the standard normal distribution .

expect

$$E(x)=\mu$$

variance

$$D(X)={\sigma }^2$$

In addition to these, there are actually random variable functions in probability theory , Multi dimensional edges and conditions and combinations , There are also covariance and moments in Chapter 4 . But I won't mention these contents , If necessary, you can see blog Or textbooks .

mathematical statistics

It's swinging , Look at this directly . I'm going back to bed .