From bd50f7c315a245bc7cf496d62fde89798321299b Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Fri, 29 Oct 2021 23:48:53 +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 --- ...or-algebra-and-space-analytic-geometry.pdf | Bin 108329 -> 108374 bytes ...or-algebra-and-space-analytic-geometry.tex | 2 +- .../8-infinite-series/infinite-series.pdf | Bin 153057 -> 153080 bytes .../8-infinite-series/infinite-series.tex | 4 ++-- ...arge-numbers-and-central-limit-theorem.pdf | Bin 106443 -> 109299 bytes ...arge-numbers-and-central-limit-theorem.tex | 20 ++++++++++++++---- .../mathematical-statistics.pdf | Bin 139649 -> 142847 bytes .../mathematical-statistics.tex | 16 +++++++++++--- 8 files changed, 32 insertions(+), 10 deletions(-) diff --git a/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.pdf 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a/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.tex +++ b/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.tex @@ -60,7 +60,7 @@ \begin{itemize} \item $\vec{a}\cdot\vec{b}=(a_x,a_y,a_z)\cdot(b_x,b_y,b_z)=a_xb_x+a_yb_y+a_zb_z$。 - \item $\vec{a}\cdot\vec{b}=\vert a\vert\vert b\vert\cos\theta$,则$\cos\theta=\dfrac{a\cdot b}{\vert a\vert\vert b\vert}=\dfrac{a_xb_x+a_yb_y+a_zb_z}{\sqrt{a_x^2+a_y^2+a_z^2}\cdot\sqrt{b_x^2+b_y^2+b_z^2}}$,其中$\theta$为$\vec{a},\vec{b}$夹角。 + \item $\vec{a}\cdot\vec{b}=\vert\vec{a}\vert\vert\vec{b}\vert\cos\theta$,则$\cos\theta=\dfrac{\vec{a}\cdot\vec{b}}{\vert a\vert\vert b\vert}=\dfrac{a_xb_x+a_yb_y+a_zb_z}{\sqrt{a_x^2+a_y^2+a_z^2}\cdot\sqrt{b_x^2+b_y^2+b_z^2}}$,其中$\theta$为$\vec{a},\vec{b}$夹角。 \item $a\bot b\Leftrightarrow\theta=\dfrac{\pi}{2}\Leftrightarrow a\cdot b=\vert a\vert\vert b\vert\cos\theta=0\Leftrightarrow a_xb_x+a_yb_y+a_zb_z=0$。 \item $Prj_ba=\dfrac{a\cdot b}{\vert b\vert}=\dfrac{a_xb_x+a_yb_y+a_zb_z}{\sqrt{b_x^2+b_y^2+b_z^2}}$称$a$在$b$上的\textbf{投影}。 \end{itemize} diff --git a/advanced-math/knowledge/8-infinite-series/infinite-series.pdf b/advanced-math/knowledge/8-infinite-series/infinite-series.pdf index 184a882100f0f038abc29a2961877bc5a8df7c34..e59202181dbacdd06cc380cfde819477325e7660 100644 GIT binary patch delta 88375 zcmZ5`LzFH&wB@Z^Rkv*0wr$(CZSyPJwr$(CZQFMB-*42fCs~;#YoFvK8~c?IZ

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z%*-P0d#l6Pf-y9@Y(}=THPHiuMe$?m7}~WHac7Kd<2{#X#Eqkzn+)a%+{G(Nc{W88m<+ZZ3w6wMnl9v4cAu^CEkdy#nzW<#pa?L2Kn^+J>oqG(K z4a2NXH8~sNnGW~=1W-B>QxdAapN@|mP)yx#ol5JdHbHxQBiEf+NVzz4yXh4C=m+~9 z@jFOS0S1$#SDlF~sK!EX8fjrQe!kmRa&Gm}BUfv!z>z=~*%9Sr9E)7RS7-KY){9EOr|U0$,$\lim\limits_{n\to\infty}P\left\{\left\vert\dfrac{1}{n}\sum\limits_{i=1}^nX_i-\dfrac{1}{n}\sum\limits_{i=1}^nEX_i\right\vert<\epsilon\right\}=1$,$\{X_n\}$服从大数定律:$\dfrac{1}{n}\sum\limits_{i=1}^nX_i\overset{P}{\longrightarrow}\dfrac{1}{n}\sum\limits_{i=1}^nEX_i$。即$\overline{X}\overset{P}{\rightarrow}E\overline{X}$。 + +即均值收敛于期望的均值。 + +证明:$E\left(\dfrac{1}{n}\sum\limits_{i=1}^nX_i\right)=\dfrac{1}{n}\sum\limits_{i=1}^nEX_i$,又$X_i$不相关,则$Cov(X_i,X_j)=0$。 + +则$D\left(\dfrac{1}{n}\sum\limits_{i=1}^nX_i\right)=\dfrac{1}{n^2}\sum\limits_{i=1}^nDX_i\leqslant\dfrac{nC}{n^2}=\dfrac{C}{n}$。 + +又根据切比雪夫不等式$1\geqslant\lim\limits_{n\to\infty}P\left\{\left\vert\dfrac{1}{n}\sum\limits_{i=1}^nX_i-\dfrac{1}{n}\sum\limits_{i=1}^nEX_i\right\vert<\epsilon\right\}\geqslant1-\dfrac{D\left(\dfrac{1}{n}\sum_{i=1}^nX_i\right)}{\epsilon^2}\geqslant1-\dfrac{C}{n\epsilon^2}\xrightarrow{n\to+\infty}1$。 + +根据夹逼定理$\lim\limits_{n\to\infty}P\left\{\left\vert\dfrac{1}{n}\sum\limits_{i=1}^nX_i-\dfrac{1}{n}\sum\limits_{i=1}^nEX_i\right\vert<\epsilon\right\}$=1。 \textbf{例题:}设$X_1,X_2,\cdots,X_n,\cdots$为相互独立的随机变量序列,$X_n$服从参数为$n$的指数分布($n\leqslant1$),则下列随机变量序列中不服从切比雪夫大数定律的是()。 @@ -78,7 +86,7 @@ $\because X_n\sim E(n)$,$\therefore EX_n=\dfrac{1}{n}$,$DX_n=\dfrac{1}{n^2}$ \subsection{伯努利大数定律} -\textcolor{violet}{\textbf{定义:}}假设$\mu_n$是$n$重伯努利试验中事件$A$发生的次数,在每次试验中事件$A$发生的概率为$p$($00$,有$\lim\limits_{n\to\infty}P\left\{\left\vert\dfrac{\mu_n}{n}-p\right\vert<\epsilon\right\}=1$。 +\textcolor{violet}{\textbf{定义:}}假设$\mu_n$是$n$重伯努利试验中事件$A$发生的次数,在每次试验中事件$A$发生的概率为$p$($00$,有$\lim\limits_{n\to\infty}P\left\{\left\vert\dfrac{\mu_n}{n}-p\right\vert<\epsilon\right\}=\lim\limits_{n\to\infty}P\left\{\left\vert\dfrac{1}{n}\sum\limits_{i=1}^nX_i-\dfrac{1}{n}\sum\limits_{i=1}^nEX_i\right\vert<\epsilon\right\}=\lim\limits_{n\to\infty}P\\\left\{\left\vert\dfrac{1}{n}\sum\limits_{i=1}^nX_i-\dfrac{1}{n}\sum\limits_{i=1}^nEX_i\right\vert<\epsilon\right\}=1$。 可以看作通过$n$重伯努利试验,一个事件的试验概率$\dfrac{\mu_n}{n}$会逼近一个固定的事件概率$p$。 @@ -92,8 +100,12 @@ $\because X_n\sim E(n)$,$\therefore EX_n=\dfrac{1}{n}$,$DX_n=\dfrac{1}{n^2}$ \subsection{辛钦大数定律} +辛钦大数定律类似切比雪夫大数定律的特殊化,将序列约束为同分布。(但是对方差没有要求,所以不能按切比雪夫定律的证明来做) + \textcolor{violet}{\textbf{定义:}}假设随机变量序列$\{X_n\}$($n=1,2,3\cdots$)是\textbf{相互独立}的\textbf{同分布}的,如果$EX_i=\mu$($i=1,2,\cdots$)\textbf{存在},则$\dfrac{1}{n}\sum\limits_{i=1}^nX_i\overset{P}{\longrightarrow}\mu$,即对任意的$\epsilon>0$有$\lim\limits_{n\to\infty}P\left\{\left\vert\dfrac{1}{n}\sum\limits_{i=1}^nX_i-\mu\right\vert<\epsilon\right\}=1$。也可以转换为即$\overline{X}\overset{P}{\rightarrow}E\overline{X}$。\medskip +即能用平均数可以来逼近期望。 + \textbf{例题:}假设随机变量序列$X_1,X_2,\cdots,X_n,\cdots$相互独立,根据辛钦大数定律,当$n\to\infty$时,$\dfrac{1}{n}\sum\limits_{i=1}^nX_i$依概率收敛于数学期望,只要$\{X_n\}$()。 $A.$有相同的数学期望\qquad$B.$服从同一离散型分布 diff --git a/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.pdf b/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.pdf index ec20f51130b7d18328d0d8abb3502e5ee688a2b5..e6ac91b2e71519dacc5195f2b7f72d34e9804144 100644 GIT binary patch delta 124353 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b/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.tex @@ -217,7 +217,7 @@ $\therefore P\{F>F_\alpha(n_2,n_1)\}=\alpha$,$P\{F\leqslant F_\alpha(n_2,n_1)\ \begin{enumerate} \item $\overline{X}\sim N\left(\mu,\dfrac{\sigma^2}{n}\right)$,即$\dfrac{\overline{X}-\mu}{\sigma/\sqrt{n}}=\dfrac{\sqrt{n}(\overline{X}-\mu)}{\sigma}\sim N(0,1)$。 - \item $\dfrac{1}{\sigma^2}\sum\limits_{i=1}^n(X_i-\mu)^2\sim\chi^2(n)$。 + \item $\dfrac{1}{\sigma^2}\sum\limits_{i=1}^n(X_i-\mu)^2\sim\chi^2(n)$。$\dfrac{(n-1)S^2}{\sigma^2}\sim\chi^2(n-1)$。 \item $\dfrac{(n-1)S^2}{\sigma^2}=\sum\limits_{i=1}^n\left(\dfrac{X_i-\overline{X}}{\sigma}\right)^2\sim\chi^2(n-1)$($\mu$未知时,在2中用$\overline{X}$代替$\mu$)。 \item $\overline{X}$与$S^2$相互独立,$\dfrac{\sqrt{n}(\overline{X}-\mu)}{S}\sim t(n-1)$($\sigma$未知时在1中用$S$代替$\sigma$)。进一步有$\dfrac{n(\overline{X}-\mu)^2}{S^2}\sim F(1,n-1)$。 \end{enumerate} @@ -236,6 +236,14 @@ $\therefore P\{F>F_\alpha(n_2,n_1)\}=\alpha$,$P\{F\leqslant F_\alpha(n_2,n_1)\ \subsubsection{矩估计法} +使用替换思想,用样本距来估计总体距。 + +\begin{enumerate} + \item 写出总体$k$阶矩$\mu_k=E(X^k)$,其中$\mu$含有参数$\theta$。 + \item 写出样本$k$阶矩$A_k=\dfrac{1}{n}\sum\limits_{i=1}^nX^k$,$A_k$只与样本有关。 + \item 令总体$k$阶矩=样本$k$阶矩,即$E(X^k)=\dfrac{1}{n}\sum\limits_{i=1}^nX^k$,就得到了$\theta$的方程。 +\end{enumerate} + \textbf{例题:}来自总体的$X$的简单随机样本$X_1,X_2,\cdots,X_n$,总体$X$的概率分布为$X\sim\left(\begin{array}{ccc} -1 & 0 & 2 \\ 2\theta & \theta & 1-3\theta @@ -329,7 +337,7 @@ $ED=\sigma^2=kE(\sum\lim\limits_{i=1}^{n-1}(X_{i+1}-X_i)^2)=kE(\sum\limits_{i=1} \subsubsection{有效性} -也称为最小方差性。 +也称为最小方差性。只有同样的无偏性才能比较有效性。 \textcolor{violet}{\textbf{定义:}}设$\hat{\theta_1}=\hat{\theta_1}(X_1,X_2,\cdots,X_n)$与$\hat{\theta_2}=\hat{\theta_2}(X_1,X_2,\cdots,X_n)$都是$\theta$的无偏估计量,若$D(\hat{\theta_1})

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线性性质:若级数$\sum\limits_{n=1}^\infty u_n$,$\sum\limits_{n=1}^\infty v_n$均收敛,且其和分别为$S$,$T$,则任给常数$a,b$,有$\sum\limits_{n=1}^\infty(au_n+bv_n)$也收敛,且其和为$aS+bT$,即$\sum\limits_{n=1}^\infty(au_n+bv_n)=a\sum\limits_{n=1}^\infty u_n+b\sum\limits_{n=1}^\infty v_n$。 \item 若级数$\sum\limits_{n=1}^\infty u_n$收敛,则其任意$m$项后余项$\sum\limits_{n=m+1}^\infty u_n$也收敛;若存在$m$项后余项$\sum\limits_{n=m+1}^\infty u_n$收敛,则$\sum\limits_{n=1}^\infty u_n$也收敛。 - \item 级数收敛必要条件:若级数$\sum\limits_{n=1}^\infty u_n$首先,则$\lim\limits_{n\to\infty}u_n=0$。 + \item 级数收敛必要条件:若级数$\sum\limits_{n=1}^\infty u_n$收敛,则$\lim\limits_{n\to\infty}u_n=0$。 \end{enumerate} 证明性质三:$u_n=S_n-S_{n-1}$,所以$\lim\limits_{n\to\infty}=\lim\limits_{n\to\infty}(S_n-S_{n-1})=\lim\limits_{n\to\infty}S_n-\lim\limits_{n\to\infty}S_{n-1}=S-S=0$。极限为0不一定收敛。 @@ -87,7 +87,7 @@ $S_n=u_1+u_2+\cdots+u_n$称为级数的\textbf{部分和},$\{S_n\}$是级数 基本就是使用放缩法判断是否有界。 -\textcolor{aqua}{\textbf{定理:}}等比级数(几何级数):$\sum\limits_{n=21}^\infty\dfrac{1}aq^{n-1}\left\{\begin{array}{l} +\textcolor{aqua}{\textbf{定理:}}等比级数(几何级数):$\sum\limits_{n=1}^\infty\dfrac{1}aq^{n-1}\left\{\begin{array}{l} \vert q\vert<1, \text{收敛} \\ \vert q\vert\geqslant 1, \text{发散} \end{array}\right.$。 diff --git a/probability-theory-and-mathematical-statistics/knowledge/4-law-of-large-numbers-and-central-limit-theorem/law-of-large-numbers-and-central-limit-theorem.pdf b/probability-theory-and-mathematical-statistics/knowledge/4-law-of-large-numbers-and-central-limit-theorem/law-of-large-numbers-and-central-limit-theorem.pdf index e64edb8010324f099d0aac8fe61b21374485c45e..6839b3ec0fa72f9d4237811fb45b48fe92f80585 100644 GIT binary patch delta 94650 zcmYhBLvWxC({^Lqn%K56NhY?POl+HXoQZAQwrx#p+tx(i^Z%=N(bc=I>Z;zH)7NA# z)WZ@~T{jrO&cP}u2{>~+@YZGL3&XqqaXuS9$PEMpSOo)2SgM-XFl(!|8^sw1K3 z$xKqtVhF2%8LLTu;qo!gWLpCZ7bSBxQph#ZCFZ6&&+c_F!di@7DuT zt{N%uAu;iu)NZ;OUqW_rW6;Y;kHiNz5oc`3BFsNz*BV=u#lXM(+biepjjM*xx=Wf@ zMBOv)f?)>DF)D`_C}Nyzi&pPns{#T3TkyyrC`wOf1T|U&M2YD{A$wOH#5kCGN%<|2# 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