From 4ac593c4c1184db29414c18f09f657a92daca917 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Sun, 28 Aug 2022 22:57:03 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...efinite-integral-and-definite-integral.pdf | Bin 208292 -> 210347 bytes ...efinite-integral-and-definite-integral.tex | 46 +++++++--- linear-algebra/exercise/5-similar/similar.pdf | Bin 90856 -> 91899 bytes linear-algebra/exercise/5-similar/similar.tex | 22 ++++- .../knowledge/5-similar/similar.pdf | Bin 100829 -> 101431 bytes .../knowledge/5-similar/similar.tex | 4 + .../random-variables-and-distribution.pdf | Bin 128171 -> 140693 bytes .../random-variables-and-distribution.tex | 82 +++++++++++++++++- .../3-digital-features/digital-features.pdf | Bin 71457 -> 91317 bytes .../3-digital-features/digital-features.tex | 50 ++++++++++- .../random-variables-and-distribution.pdf | Bin 189158 -> 189645 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$A^{-1}\xi=\dfrac{1}{\lambda}\xi$。 + \item $A^*\alpha=\dfrac{\vert A\vert}{\lambda}\xi$。 \end{itemize} \subsubsection{特征向量性质} @@ -67,6 +70,7 @@ \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。 + \item 若$A$可逆,则$A$、$A^{-1}$、$A^*$的特征向量相同。 \end{itemize} 证明性质二:利用定义法,首先$A\xi_1=\lambda_1\xi_1$,$A\xi_2=\lambda_2\xi_2$。 diff --git a/probability-theory-and-mathematical-statistics/exercise/2-random-variables-and-distribution/random-variables-and-distribution.pdf b/probability-theory-and-mathematical-statistics/exercise/2-random-variables-and-distribution/random-variables-and-distribution.pdf index b2b9ebf575faea039704aea7d0509993ff6d3eb3..c9c3949b44113cf659e883ccdfd5a86732f301b3 100644 GIT binary patch delta 134847 zcmZs>V~j2g)Fjxpt=qP3+kM-%ZR2U%wr$(CZQI?qjd`=#-F%aof0dl%oIiDvs!CN| 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xD@71s>$m6)Z|-3BzhUfuJx3Fpe?e_y7dI~#a|?JDZY~~fW_St;aYYIE{{k!(u-E_q diff --git a/probability-theory-and-mathematical-statistics/knowledge/2-random-variables-and-distribution/random-variables-and-distribution.tex b/probability-theory-and-mathematical-statistics/knowledge/2-random-variables-and-distribution/random-variables-and-distribution.tex index 894641b..c23419d 100644 --- a/probability-theory-and-mathematical-statistics/knowledge/2-random-variables-and-distribution/random-variables-and-distribution.tex +++ b/probability-theory-and-mathematical-statistics/knowledge/2-random-variables-and-distribution/random-variables-and-distribution.tex @@ -263,7 +263,7 @@ $P\{X=x_i\}=P\{X\leqslant x_i\}-P\{Xs+t|X>s\}=P\{X>t\}$。 +无记忆性\textcolor{aqua}{\textbf{定理:}}若$X\sim E(\lambda)$,则$P\{X>s+t|X>s\}=P\{X>t\}=e^{-\lambda t}$。 即在指数分布下事情发生的概率与前面所经过的时间无关,如果$T$是某一元件的寿命,已知元件使用了$t$小时,它总共使用至少$s+t$小时的条件概率,与从开始使用时算起它使用至少$s$小时的概率相等。 @@ -503,12 +503,16 @@ $p_{\cdot j}=P\{Y=y_i\}=\sum\limits_{i=1}^\infty P\{X=x_i,Y=y_j\}=\sum\limits_{i \textcolor{violet}{\textbf{定义:}}设$(X,Y)\sim f(x,y)$,则$X$的边缘分布函数为$P\{X\leqslant x\}=F_X(x)=F(-\infty,x)=\int_{-\infty}^x\left[\int_{-\infty}^{+\infty}f(u,v)\,\textrm{d}v\right]\textrm{d}u$,所以$X$为连续型随机变量,其概率密度$f_X(x)=\int_{-\infty}^{+\infty}f(x,y)\,\textrm{d}y$,称$f_X(x)$为$(X,Y)$关于$X$的\textbf{边缘概率密度}。同理$Y$也为连续型随机变量,关于$Y$的边缘分布函数为$P\{Y\leqslant y\}=F_Y(y)=\int_{-\infty}^y[\int_{-\infty}^{+\infty}f(x,y)\,\textrm{d}x]\textrm{d}y$,其概率密度为$f_Y(y)=\int_{-\infty}^{+\infty}f(x,y)\,\textrm{d}x$。 +联合概率密度决定边缘概率密度,所以相同联合概率密度的拥有同样边缘概率密度。 + \subsection{条件概率密度} \textcolor{violet}{\textbf{定义:}}设$(X,Y)\sim f(x,y)$,边缘概率密度$f_X(x)>0$,则称$f_{Y|X}(y|x)=\dfrac{f(x,y)}{f_X(x)}$为$Y$在$X=x$条件下的\textbf{条件概率密度}。同理$X$在$Y=y$条件下的条件概率密度为$f_{X|Y}(x|y)=\dfrac{f(x,y)}{f_Y(y)}$。 若$f_X(x)>0$,$f_Y(y)>0$,则有概率密度乘法公式$f(x,y)=f_X(x)f_{Y|X}(y|x)=f_Y(y)f_{X|Y}(x|y)$。 +如果$XY$独立,则$f(x,y)=f(x)f(y)$,此时条件概率密度就等于边缘概率密度。 + \textcolor{violet}{\textbf{定义:}}$Y$在$X=x$条件下的\textbf{条件分布函数}为$F_{Y|X}(y|x)=\int_{-\infty}^yf_{Y|X}(y|x)\,\textrm{d}y=\displaystyle{\int_{-\infty}^y\dfrac{f(x,y)}{f_X(x)}\textrm{d}y}$,同理$X$在$Y=y$条件下的条件分布函数为$F_{X|Y}(x|y)=$\\$\int_{-\infty}^xf_{X|Y}(x|y)\,\textrm{d}x=\displaystyle{\int_{-\infty}^x\dfrac{f(x,y)}{f_Y(y)}}\textrm{d}x$。 \subsection{二维均匀分布} @@ -531,7 +535,7 @@ $p_{\cdot j}=P\{Y=y_i\}=\sum\limits_{i=1}^\infty P\{X=x_i,Y=y_j\}=\sum\limits_{i \begin{itemize} \item $X\sim N(\mu_1,\sigma_1^2)$,$Y\sim N(\mu_2,\sigma_2^2)$,$\rho$为$X$与$Y$的相关系数,即$\rho=\dfrac{Cov(X,Y)}{\sqrt{DX}\sqrt{DY}}=\dfrac{Cov(X,Y)}{\sigma_1\sigma_2}$。 \item $X,Y$的条件分布都是正态分布。 - \item $aX+bY$($a\neq0$或$b\neq0$)服从正态分布。 + \item $aX\pm bY$($a\neq0$或$b\neq0$)服从正态分布$N(a\mu_1\pm b\mu_2,a^2\sigma_1^2+b^2\sigma_2^2)$。 \item $XY$相互独立的充要条件是$XY$不相关,即$\rho=0$。 \end{itemize}

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a/probability-theory-and-mathematical-statistics/exercise/2-random-variables-and-distribution/random-variables-and-distribution.tex b/probability-theory-and-mathematical-statistics/exercise/2-random-variables-and-distribution/random-variables-and-distribution.tex index 322251e..c918018 100644 --- a/probability-theory-and-mathematical-statistics/exercise/2-random-variables-and-distribution/random-variables-and-distribution.tex +++ b/probability-theory-and-mathematical-statistics/exercise/2-random-variables-and-distribution/random-variables-and-distribution.tex @@ -41,6 +41,74 @@ 分布函数变量区域左闭右开,概率密度则不要求。 +\section{分布概念} + +\subsection{分布函数存在性} + +分布函数的充要条件是: + +\begin{enumerate} + \item $F(x)$单调不减。 + \item $\lim\limits_{x\to-\infty}F(x)=0$。 + \item $\lim\limits_{x\to+\infty}F(x)=1$。 + \item $F(x)$右连续。 +\end{enumerate} + +这也是下面的计算需要考虑的性质。 + +\textbf{例题:}设随机变量$X$的分布函数$F(x)$,判断$F(ax)$、$F(x^2+1)$、$F(x^3-1)$、$F(\vert X\vert)$是否可以做出分布函数。 + +解:逐个分析$F(g(x))$中的$g(x)$走向。 + +$\lim\limits_{x\to+\infty}F(ax)=a\lim\limits_{x\to+\infty}F(x)=a$,不满足。 + +由于$x^2+1$在$x\to-\infty$的趋向为$+\infty$而不是$-\infty$,$\lim\limits_{x\to-\infty}F(x^2+1)=\lim\limits_{x\to+\infty}F(x^2+1)=\lim\limits_{x\to-\infty}F(x)=1\neq0$,不满足。 + +$F(x^3-1)$满足分布函数的四个条件,$g(x)=x^3-1$在两边趋向与原函数一致。 + +$F(\vert x\vert)$的$g(x)=\vert x\vert$在$x\to-\infty$的趋向为$+\infty$而不是$-\infty$,不满足$\lim\limits_{x\to-\infty}F(x)=0$,不满足。 + +\subsection{分布函数计算} + +这里所说的分布函数都是不定式,需要利用分布函数性质来换算。 + +\textbf{例题:}随机变量$X$的概率密度函数$f(x)$满足$f(1+x)=f(1-x)$,且$\int_1^2f(x)\,\textrm{d}x=0.3$,则求分布函数$F(0)$的值。 + +解:已知$f(1+x)=f(1-x)$,所以$f(x)$关于$x=1$对称。所以$\int_1^{+\infty}f(x)\,\textrm{d}x=\int_1^{-\infty}f(x)\,\textrm{d}x=\dfrac{1}{2}$。 + +所以$\int_1^2f(x)\,\textrm{d}x=\int_0^1f(x)\,\textrm{d}x=0.3$。 + +$F(0)=\int_{-\infty}^0f(x)\,\textrm{d}x=1-\int_0^{+\infty}f(x)\,\textrm{d}x=1-\int_1^{+\infty}f(x)\,\textrm{d}x-\int_0^1f(x)\,\textrm{d}x=0.2$。 + +\subsection{概率密度存在性} + +分布函数的充要条件是: + +\begin{enumerate} + \item $f(x)\geqslant0$。 + \item $\int_{-\infty}^{+\infty}f(x)\,\textrm{d}x=1$。 +\end{enumerate} + +\textbf{例题:}随机变量$X$的概率密度函数$f(x)$,判断$f(2x)$、$f(2-x)$是否可以作出概率密度函数。 + +解:$\int_{-\infty}^{+\infty}f(2x)\,\textrm{d}x=\dfrac{1}{2}\int_{-\infty}^{+\infty}f(2x)\,\textrm{d}(2x)=\dfrac{1}{2}\neq1$,不满足。 + +$\int_{-\infty}^{+\infty}f(2-x)\,\textrm{d}x=-\int_{-\infty}^{+\infty}f(2-x)\,\textrm{d}(2-x)$,令$2-x=t$,则$x\to-\infty$时$t\to+\infty$,$x\to+\infty$时$t\to-\infty$,$=-\int_{+\infty}^{-\infty}f(t)\,\textrm{d}t=\int_{-\infty}^{+\infty}f(t)\,\textrm{d}t=1$。 + +\subsection{概率密度计算} + +\textbf{例题:}随机变量$X$的分布函数和概率密度函数为$F(x)$,$f(x)$,求$-X$的分布函数和概率密度。 + +解:已知$F(x)=P\{X\leqslant x\}$,则$F'(-x)=\{-X\leqslant x\}$,即$F'(-x)=P\{X\geqslant-x\}=1-P\{X\leqslant-x\}=1-F(-x)$。 + +对$F'(-x)$进行求导:$[1-F(-x)]'=f(-x)$。 + +\textbf{例题:}随机变量$X$的概率密度为$f(x)$,求随机变量$2X+3$的概率密度函数。 + +解:设$Y=2X+3$,所以$Y$的分布函数$F_Y(y)$为$F_Y(x)=P\{Y\leqslant x\}=P\{2X+3\leqslant x\}=P\left\{X\leqslant\dfrac{x-3}{2}\right\}=\int_{-\infty}^{\frac{t-3}{2}}f(x)\,\textrm{d}x$。 + +所以其概率密度为$f_Y(x)=(\int_{-\infty}^{\frac{x-3}{2}}f(x)\,\textrm{d}x)'=\dfrac{1}{2}f\left(\dfrac{x-3}{2}\right)$。 + \section{一维随机变量} \subsection{一维随机变量分布} @@ -226,9 +294,21 @@ $\because$面积为$\alpha$的下标为$\alpha$,$\therefore$面积为$\dfrac{1 \subsection{二维连续型随机变量} +有两种主要方法,一个是分布函数法,一个是卷积公式法。如果给出联合概率密度使用分布函数法,如果给出边缘概率密度使用卷积公式法。 + \subsubsection{联合概率} -\paragraph{概率函数} \leavevmode \medskip +\paragraph{联合概率密度} \leavevmode \medskip + +利用联合概率函数来计算,有两种方式,一种是以概率形式,一种是以积分形式。 + +\textbf{例题:}已知二维随机变量$(X_1,X_2)$的概率密度函数$f_1(x_1,x_2)$,求$Y_1=2X_1$,$Y_2=\dfrac{1}{3}X_2$的联合概率密度$f_2(y_1,y_2)$。 + +解:概率形式:设$F_2(y_1,y_2)=P\{Y_1\leqslant y_1,Y_2\leqslant y_2\}=P\{2X_1\leqslant y_1,\dfrac{1}{3}X_2\leqslant y_2\}=P\{X_1\leqslant\dfrac{y_1}{2},X_2\leqslant3y_2\}=F_1(\dfrac{y_1}{2},3y_2)$。所以$f_2(y_1,y_2)=\dfrac{3}{2}f_1(\dfrac{y_1}{2},3y_2)$。 + +积分形式:已知$\int_{-\infty}^{+\infty}\textrm{d}x_1\int_{-\infty}^{+\infty}f(x_1,x_2)\,\textrm{d}x_2=1$,将$x_1=\dfrac{y_1}{2}$,$x_2=3y_2$代入$\int_{-\infty}^{+\infty}\textrm{d}(\dfrac{y_1}{2})\int_{-\infty}^{+\infty}f(\dfrac{y_1}{2},3y_2)\,\textrm{d}(3y_2)=1$,$\dfrac{3}{2}\int_{-\infty}^{+\infty}\textrm{d}y_1\int_{-\infty}^{+\infty}f(\dfrac{y_1}{2},3y_2)\,\textrm{d}y_2=1$,所以$f_2(y_1,y_2)=\dfrac{3}{2}f_1(\dfrac{y_1}{2},3y_2)$。 + +\paragraph{联合概率函数} \leavevmode \medskip 已知联合概率密度,可以求概率函数,通过二重积分的方式,图像面积即是概率。 diff --git a/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.pdf b/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.pdf index 1d209be34d0f55bdfbefb9e7ba1f26a15a9a7e16..dc3593375d590f5916f225dd92de7c13f7a9046b 100644 GIT binary patch delta 88593 zcmZs?Q*C82h6`6 zn{mSZiOCPePZ+Wo$cG$4rF@R??TgjR1+@CEVRNSlIPs%Y<6Pv3Xd8N-L-@r23Atup zae>rrOM~2jAYJiAJwB%1`1<&|%(!9V<)^kQ1snxcsj=Lyji%aKkO=066l=WL)(bmd 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