diff --git a/linear-algebra/exercise/5-similar/similar.pdf b/linear-algebra/exercise/5-similar/similar.pdf index bf89307..449e262 100644 Binary files a/linear-algebra/exercise/5-similar/similar.pdf and b/linear-algebra/exercise/5-similar/similar.pdf differ diff --git a/linear-algebra/exercise/5-similar/similar.tex b/linear-algebra/exercise/5-similar/similar.tex index 690bca9..6d5571f 100644 --- a/linear-algebra/exercise/5-similar/similar.tex +++ b/linear-algebra/exercise/5-similar/similar.tex @@ -39,6 +39,8 @@ \section{特征值与特征向量} +首先根据$\vert\lambda E-A\vert=0$求出$\lambda$，然后把$\lambda$逐个带入$(\lambda E-A)x=0$，根据齐次方程求解方法进行初等变换求出基础解系。这个基础解系就是当前特征值的特征向量。 + \subsection{迹} \textbf{例题：}已知$A$是3阶方阵，特征值为1，2，3，求$\vert A\vert$的元素$a_{11},a_{22},a_{33}$的代数余子式$A_{11},A_{22},A_{33}$的和$\sum\limits_{i=1}^3A_{ii}$。 diff --git a/linear-algebra/knowledge/5-similar/similar.pdf b/linear-algebra/knowledge/5-similar/similar.pdf index 99f3c5e..a9dfa29 100644 Binary files a/linear-algebra/knowledge/5-similar/similar.pdf and b/linear-algebra/knowledge/5-similar/similar.pdf differ diff --git a/linear-algebra/knowledge/5-similar/similar.tex b/linear-algebra/knowledge/5-similar/similar.tex index 73b81da..588b3a2 100644 --- a/linear-algebra/knowledge/5-similar/similar.tex +++ b/linear-algebra/knowledge/5-similar/similar.tex @@ -64,7 +64,7 @@ \subsubsection{特征向量性质} \begin{itemize} - \item $k$重特征值$\lambda$至多只有$k$个线性无关的特征向量。 + \item $k$重特征值$\lambda$至多只有$k$个线性无关的特征向量。一共有$k$个特征向量。 \item 若$\xi_1$和$\xi_2$是$A$的属于不同特征值$\lambda_1$和$\lambda_2$的特征向量，则$\xi_1$和$\xi_2$线性无关。 \item 若$\xi_1$和$\xi_2$是$A$的属于同特征值$\lambda$的特征向量，则$k_1\xi_1+k_2\xi_2$（$k_1k_2$不同时为0）仍是$A$的属于特征值$\lambda$的特征向量。 \end{itemize} @@ -88,15 +88,23 @@ $Ak_1\xi_1+Ak_2\xi_2=k_1\lambda_1\xi_1+k_2\lambda_1\xi_2=0$。又$k_1\xi_1+k_2\x 性质一是因为特征向量的性质而来。从几何来理解，特征向量表示的是矩阵变换中只有伸缩变换没有旋转变换的方向向量，特征值是这个方向的伸缩系数，一个方向当然只有一个伸缩系数。 -\subsection{运算} +\subsubsection{运算性质} $\because\lambda\xi-A\xi=0$，$\therefore(\lambda E-A)\xi=0$，又$\xi\neq0$，$\therefore(\lambda E-A)x=0$有非零解。 从而$\lambda E-A$所表示的方阵线性相关，为降秩，从而$\vert\lambda E-A\vert=0$。 +其中$n-r(\lambda E-A)$的值就是特征向量中自由变量的个数。 + $\vert\lambda E-A\vert=0$也称为特征方程或是特征多项式，解出的$\lambda_i$就是特征值。 -将$\lambda_i$代回原方程，根据极大线性无关组解出通解就是$\xi$。 +将$\lambda_i$代回原方程求解。即$(\lambda E-A)x=0$有非零解，齐次方程只有唯一零解和无穷非零解两种结果，所以这里求出来的就是无穷非零解，所以只用求出解的基础解系即可。 + +根据极大线性无关组解出通解就是$\xi$，非线性无关组的变量设为自由变量（不能被约束的）用来表示其他变量。 + +如果没有行阶梯型，则对于一列全是0的变量就是自由变量。 + +\subsection{运算} \subsubsection{具体型} diff --git a/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.pdf b/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.pdf index 97c2416..1d209be 100644 Binary files a/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.pdf and b/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.pdf differ diff --git a/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.tex b/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.tex index 7e67c53..cb7d187 100644 --- a/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.tex +++ b/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.tex @@ -100,4 +100,42 @@ $Cov(X,Y)=E(XY)-E(X)E(Y)$。 解：$\because Y_1=\sum\limits_{i=2}^nX_i$，$Y_2=\sum\limits_{j=1}^{n-1}X_j$，$DX_i=\sigma^2$。 +\section{独立性与相关性} + +独立范围小于不相关范围。所以我们一般先用数字特征判断相关性再用分布判断独立性。 + +$Cov(X,Y)=E(XY)-EXEY\left\{\begin{array}{l} + \neq0\Leftrightarrow XY\text{相关}\Rightarrow X\text{与}Y\text{不独立} \\ + =0\Leftrightarrow XY\text{不相关，分布}\left\{\begin{array}{l} + XY\text{独立} \\ + XY\text{不独立} \\ + \end{array}\right. +\end{array}\right.$ + +且如果服从二维正态分布，则$XY$独立与不相关等价。 + +\subsection{独立性} + +通过分布来确定独立性。如独立条件是$f(x,y)=f_X(x)f_Y(y)$，$P\{X=x_i,Y=y_j\}=P\{X=x_i\}P\{Y=y_j\}$。 + +\subsection{相关性} + +通过数字特征来判断相关性。如不相关性条件是$\rho_{XY}=0$、$Cov(X,Y)=0$、$E(XY)=EXEY$、$D(X\pm Y)=DX+DY$。 + +\section{切比雪夫不等式} + +切比雪夫不等式用于估算随机变量在区间的概率，证明收敛性问题。 + +\subsection{区间概率} + +常用变式$P\{\vert Z-EZ\vert\geqslant\epsilon\}\leqslant\dfrac{DZ}{\epsilon^2}$或$P\{\vert Z-EZ\vert<\epsilon\}\geqslant1-\dfrac{DZ}{\epsilon^2}$，$Z=f(X)$。 + +\textbf{例题：}已知随机变量$XY$，$EX=EY=2$、$DX=1$、$DY=4$，$\rho_{XY}=0.5$，估计概率$P\{\vert X-Y\vert\geqslant6\}$。 + +解：已知$\rho_{XY}=0.5=\dfrac{Cov(X,Y)}{\sqrt{DX}\sqrt{DY}}=\dfrac{Cov(X,Y)}{2}$，$Cov(X,Y)=1=E(XY)-EXEY$，$E(XY)=5$。 + +令$X-Y=Z$，$EZ=EX-EY=0$，$DZ=DX+DY-2Cov(X,Y)=1+4-2=3$。 + +取$\epsilon=6$，由切比雪夫不等式得$P\{\vert X-Y\vert\geqslant6\}=P\{\vert Z-0\vert\geqslant6\}\leqslant\dfrac{DZ}{\epsilon^2}=\dfrac{3}{6^2}=\dfrac{1}{12}$。 + \end{document} diff --git a/probability-theory-and-mathematical-statistics/knowledge/3-digital-features/digital-features.pdf b/probability-theory-and-mathematical-statistics/knowledge/3-digital-features/digital-features.pdf index 3eca5a3..813c555 100644 Binary files a/probability-theory-and-mathematical-statistics/knowledge/3-digital-features/digital-features.pdf and b/probability-theory-and-mathematical-statistics/knowledge/3-digital-features/digital-features.pdf differ diff --git a/probability-theory-and-mathematical-statistics/knowledge/3-digital-features/digital-features.tex b/probability-theory-and-mathematical-statistics/knowledge/3-digital-features/digital-features.tex index 1a56bef..7d72d27 100644 --- a/probability-theory-and-mathematical-statistics/knowledge/3-digital-features/digital-features.tex +++ b/probability-theory-and-mathematical-statistics/knowledge/3-digital-features/digital-features.tex @@ -87,7 +87,7 @@ \subsection{切比雪夫不等式} -\textcolor{violet}{\textbf{定义：}}若随机变量$X$的方差$DX$存在，则对任意$\epsilon>0$，有$P\{\vert X-EX\vert\leqslant\epsilon\}\leqslant\dfrac{DX}{\epsilon^2}$或$P\{\vert X-EX\vert<\epsilon\}\geqslant1-\dfrac{DX}{\epsilon^2}$。 +\textcolor{violet}{\textbf{定义：}}若随机变量$X$的方差$DX$存在，则对任意$\epsilon>0$，有$P\{\vert X-EX\vert\geqslant\epsilon\}\leqslant\dfrac{DX}{\epsilon^2}$或$P\{\vert X-EX\vert<\epsilon\}\geqslant1-\dfrac{DX}{\epsilon^2}$。 $P\{\vert X-EX\vert\geqslant\epsilon\}$即代表变量与期望的差距大于某个值的概率，$DX$就是方差，$DX$越小证明波动越小，波动在$\epsilon$外的概率就越小，反之同理，而$\epsilon$越小，则$\dfrac{DX}{\epsilon^2}$越大，则代表$X$靠近期望$EX$的概率越大，反之同理。 diff --git a/probability-theory-and-mathematical-statistics/knowledge/4-law-of-large-numbers-and-central-limit-theorem/law-of-large-numbers-and-central-limit-theorem.pdf b/probability-theory-and-mathematical-statistics/knowledge/4-law-of-large-numbers-and-central-limit-theorem/law-of-large-numbers-and-central-limit-theorem.pdf index 6839b3e..b8fbb74 100644 Binary files a/probability-theory-and-mathematical-statistics/knowledge/4-law-of-large-numbers-and-central-limit-theorem/law-of-large-numbers-and-central-limit-theorem.pdf and b/probability-theory-and-mathematical-statistics/knowledge/4-law-of-large-numbers-and-central-limit-theorem/law-of-large-numbers-and-central-limit-theorem.pdf differ diff --git a/probability-theory-and-mathematical-statistics/knowledge/4-law-of-large-numbers-and-central-limit-theorem/law-of-large-numbers-and-central-limit-theorem.tex b/probability-theory-and-mathematical-statistics/knowledge/4-law-of-large-numbers-and-central-limit-theorem/law-of-large-numbers-and-central-limit-theorem.tex index 8006b5f..3d6ad21 100644 --- a/probability-theory-and-mathematical-statistics/knowledge/4-law-of-large-numbers-and-central-limit-theorem/law-of-large-numbers-and-central-limit-theorem.tex +++ b/probability-theory-and-mathematical-statistics/knowledge/4-law-of-large-numbers-and-central-limit-theorem/law-of-large-numbers-and-central-limit-theorem.tex @@ -40,6 +40,10 @@ 这些定理与定律是针对极大量数据的概率分析，是概率论向数理统计的过渡。 +大数定律与中心定理成立的首要条件都是独立同分布且数学期望存在。 + +大数定律用于表示实验均值逼近期望，中心极限定理表示独立同分布的变量逼近正态分布。 + \section{依概率收敛} \textcolor{violet}{\textbf{定义：}}设随机变量$X$与随机变量序列$\{X_n\}$（$n=1,2,3\cdots$），如果对任意的$\epsilon>0$，有$\lim\limits_{n\to\infty}P\{\vert X_n-X\vert\geqslant\epsilon\}=0$或$\lim\limits_{n\to\infty}P\{\vert X_n-X\vert<\epsilon\}=1$，则称随机变量序列$\{X_n\}$\textbf{依概率收敛于随机变量$X$}，记为$\lim\limits_{n\to\infty}X_n=X(P)$或$X_n\overset{P}{\rightarrow}X(n\to\infty)$。 @@ -132,6 +136,10 @@ $C.$服从同一泊松分布\qquad$D.$服从同一连续型分布 中心极限定理总结来看均为：若$X_i$独立同分布于某一分布$F$，则$\sum\limits_{i=1}^nX_i\overset{n\to\infty}{\sim}N(n\mu,n\sigma^2)$。 +即无论什么分布的事件在次数无限大的情况下近乎正态分布。 + +$n$一般大于10以上即可使用中心极限定理。 + \subsection{列维-林德伯格定理} \textcolor{violet}{\textbf{定义：}}假设$\{X_n\}$是独立分布的随机变量序列，若$EX_i=\mu$，$DX_i=\sigma^2>0$（$i=1,2,\cdots$）存在，则对任意的实数$x$，有$\lim\limits_{n\to\infty}P\left\{\dfrac{\sum\limits_{i=1}^nX_i-n\mu}{\sqrt{n}\sigma}\leqslant x\right\}=\dfrac{1}{\sqrt{2}\pi}\int_{-\infty}^xe^{-\frac{t^2}{2}}\,\textrm{d}t=\varPhi(x)$。（正态分布标准化） @@ -152,6 +160,8 @@ $\dfrac{\sum\limits_{i=1}^nX_i-n\mu}{\sqrt{n}\sigma}\sim N(0,1)$，$\sum\limits_ \subsection{棣莫弗-拉普拉斯定理} +或简称拉普拉斯中心极限定理。是列维-林德伯格定理的特殊情况。 + \textcolor{violet}{\textbf{定义：}}假设随机变量$Y_n\sim B(n,p)$（$0