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概率论与数理统计考试重点复习路线
2022-07-05 04:17:00 【物联黄同学】
概率论与数理统计考试重点复习路线
文章目录
前言
希望能够通过一份简单的路线,实现精准高效的备战明天的考试。话不多说,冲冲冲!
内容分为概率论与数理统计两个部分,中间的串联是第五章的大数定律和中心极限定理。
MindMap
概率论部分
数理统计部分
概率论
基本概念
这个部分的内容,我的建议是直接看我之前的blog,或者看书以及其他网课ppt之类的。
关于随机变量的分布函数我不去列举,大家可以直接通过分布律或者概率密度推导
离散型
0-1 分布
X~b§
分布律
P { X = k } = p k ( 1 − p ) 1 − k , k = 1 , 0 P\{X=k \} = p^k(1-p)^{1-k}, \qquad k = 1, 0 P{ X=k}=pk(1−p)1−k,k=1,0
X | 0 | 1 |
---|---|---|
p_k | 1-p | p |
数学期望
E ( X ) = p E(X) = p E(X)=p
方差
D ( X ) = ( 1 − p ) ⋅ p D(X) = (1-p)\cdot p D(X)=(1−p)⋅p
二项分布
X~b(n, p)
分布律
P { X = k } = p k ( 1 − p ) 1 − k P\{X=k \} = p^k(1-p)^{1-k} P{ X=k}=pk(1−p)1−k
数学期望
E ( X ) = n p E(X) = np E(X)=np
方差
D ( X ) = n ( 1 − p ) ⋅ p D(X) = n(1-p)\cdot p D(X)=n(1−p)⋅p
泊松分布
X~π(λ)
分布律
P { X = k } = λ k e − λ k ! , k = 0 , 1 , 2... P\{X=k \} = \frac{\lambda^ke^{-\lambda}}{k!}, \qquad k=0,1,2... P{ X=k}=k!λke−λ,k=0,1,2...
泊松定理
就是用泊松去逼近二项,np=λ
lim n → ∞ C n k ( 1 − p n ) n − k = λ k e − λ k ! \lim_{n\rightarrow \infty}{C_n^k(1-p_n)^{n-k}} = \frac{\lambda^ke^{-\lambda}}{k!} n→∞limCnk(1−pn)n−k=k!λke−λ
数学期望
E ( X ) = λ E(X) = \lambda E(X)=λ
方差
D ( X ) = λ D(X) = \lambda D(X)=λ
连续型
均匀分布
X~U(a, b)
概率密度
KaTeX parse error: No such environment: align at position 26: …eft \{ \begin{̲a̲l̲i̲g̲n̲}̲ &\frac{1}{b…
期望
E ( X ) = a + b 2 E(X) = \frac {a+b}{2} E(X)=2a+b
方差
D ( X ) = ( b − a ) 2 12 D(X) = \frac{(b-a)^2}{12} D(X)=12(b−a)2
指数分布
X~E(θ)
概率密度
KaTeX parse error: No such environment: align at position 26: …eft \{ \begin{̲a̲l̲i̲g̲n̲}̲ &\frac{1}{\…
期望
E ( X ) = θ E(X) = \theta E(X)=θ
方差
D ( X ) = θ 2 D(X) = \theta^2 D(X)=θ2
正态分布
X~N(μ, σ)
概率密度
f ( x ) = 1 2 π σ e − ( x − u ) 2 2 σ 2 , − ∞ < x < ∞ f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-u)^2}{2\sigma^2}}, \qquad -\infty < x < \infty f(x)=2πσ1e−2σ2(x−u)2,−∞<x<∞
标准正态分布
X ∼ N ( 0 , 1 2 ) φ ( x ) = 1 2 π e − x 2 / 2 X\sim N(0, 1^2)\\ \varphi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2} X∼N(0,12)φ(x)=2π1e−x2/2
期望和方差,一般情况下只要先化成标准正态分布,然后用标准的正态分布的方差和期望求解即可。
期望
E ( x ) = μ E(x) = \mu E(x)=μ
方差
D ( X ) = σ 2 D(X) = \sigma^2 D(X)=σ2
概率论部分的除了这些其实还有像随机变量函数,多维的边缘和条件以及联合,还有第四章的协方差和矩。但是这些内容我就不提了,有需要的可以看blog或者课本。
数理统计
开摆了,这个直接看吧。我要回去睡觉了。
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