初等概率论

Bernoulli分布, Poisson分布, 正态分布可加性

1. \(X\sim B(n,p), Y\sim B(m,p)\Rightarrow X+Y\sim B(m+n,p)\)

2. \(X\sim P(\lambda), Y\sim P(\mu)\Rightarrow X+Y\sim P(\lambda+\mu)\)

3. \(X\sim N(\mu_1,\sigma_1^2),Y\sim N(\mu_2,\sigma_2^2)\Rightarrow X+Y\sim N(\mu_1+\mu_2,\sigma_1^2+\sigma_2^2)\)

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假设\((X,Y)\sim N(\mu_1,\mu_2,\sigma_1^2,\sigma_2^2,\rho)\)

1. \(X\sim N(\mu_1,\sigma_1^2),Y\sim N(\mu_2,\sigma_2^2)\)

2. 给定\(X=x\),则\(Y\sim N(\mu_2+\rho\frac{\sigma_2}{\sigma_1}(x-\mu_1),\sigma_2^2(1-\rho^2))\)

3. \(X,Y\) 独立\(\Leftrightarrow \rho=0\)

4. 给定\(a,b\in R,aX+bY\sim N(\mu,\sigma^2)\)

   \(\Rightarrow \mu = a\mu_1+b\mu_2,\sigma^2=a^2\sigma_1^2+b^2\sigma_2^2+2ab\rho\sigma_1\sigma_2\)

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\(k\) 阶矩

\(EX^k<\infty\Rightarrow EX^k\)的\(X\)的\(k\)阶矩

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\(X\sim P(\lambda)\) \(\Rightarrow\) \(EX=\lambda,EX^2=\lambda+\lambda^2\)

\(EX(X-1)\cdots(X-k+1)=\lambda^k\)

由此可以求出\(P(\lambda)\)的各阶矩

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\(X\sim N(0,\sigma^2)\) \(\Rightarrow\) \(m_{2k}=(zk-1)!!\sigma^{2k},m_{2k-1}=0\)

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矩母函数 \(G_X(t)=Ee^{tX},|t|<|t_0|:Ee^{tX}<\infty,|t|<t_0\)

\(X\sim P(\lambda)\) \(\Rightarrow\) \(G_X(t)=e^{\lambda(e^t-1)}\)

\(X\sim N(0,\sigma^2)\) \(\Rightarrow\) \(G_X(t)=e^{\frac{1}{2}\sigma^2t^2}\)

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\(Chebyshev\) 不等式

\(P(|X-\mu|\geq \varepsilon)\leq \frac{\sigma^2}{\varepsilon^2}\)

推广为:\(f:R\to R^+\)为非负单调不减函数,\(Ef(X)<\infty\),那么\(P(X>\varepsilon)\leq \frac{Ef(X)}{f(\varepsilon)}\))

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协方差和相关系数

\(Cov(X,Y)=E(X-\mu_X)(Y-\mu_Y)\)

\(\rho(X,Y)=\frac{Cov(X,Y)}{\sigma_X\sigma_Y}\)=\(\frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}\)

\((X,Y)\)的数字特征包括均值向量和协方差矩阵

\(\mu=(EX,EY)\),\(\Sigma=\begin{pmatrix}Var(X)&Cov(X,Y)\\Cov(Y,X)&Var(Y)\end{pmatrix}\)

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特征函数

\(\phi_X(t)=Ee^{itX}\)=\(\int_{-\infty}^{\infty}e^{itx}dF_X(x),t\in R\)

密度函数\(f_X(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-itx}\phi_X(t)dt\)

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各种收敛

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极限定理 1

\(\{\xi_n\}\)独立同分布,\(P(\xi_n=1)=p,P(\xi_n=0)=1-p\),令\(S_n=\sum\limits_{i=1}^n\xi_i\)那么:

1.\((Bernoulli):\frac{S_n}{n}\overset{p}\to p\)

2.\((De Moivre-Laplace):\frac{S_n-np}{\sqrt{np(1-p)}}\overset{d}\to N(0,1)\)

3.\((Borel):\frac{S_n}{n}\to p,s.e.\)

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Poisson极限定理

\(\forall n,\{\xi_{n,k}\}_{k=1}^n\)独立同分布,\(P(\xi_{n,k}=1)=p_n,P(\xi_{n,k}=0)=1-p_n,1\leq k\leq n\), 若:\(n\to\infty\)时,\(p_n\to\infty,np_n\to\lambda\in R^+\),令\(S_n=\sum\limits_{k=1}^n\xi_{n,k}\),那么:

\[S_n\overset{d}\to \mathcal{P}(\lambda)\]

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\(\{\xi_n\}\)独立同分布,令\(S_n=\sum\limits_{i=1}^n\xi_i\)那么:

1.\((Khintchine):\frac{S_n}{n}\overset{p}\to \mu,a.e.\) \(\iff\) \(E|\xi_1|<\infty,E\xi=\mu\)

2.\((Feller-Levy):\frac{S_n}{\sqrt{n\sigma^2}}\overset{d}\to N(0,1)\) \(\iff\) \(E\xi_n=0,Var\xi_n=E\xi_n^2=\sigma^2\)

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