From 2a2cbf23dd0faa8ba750aa424b10ce217e29e2f5 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Thu, 15 Jul 2021 23:11:53 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E6=95=B0=E7=90=86=E7=BB=9F?= =?UTF-8?q?=E8=AE=A1?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../random-variables-and-distribution.pdf | Bin 76386 -> 76381 bytes .../random-variables-and-distribution.tex | 2 +- .../3-digital-features/digital-features.pdf | Bin 0 -> 41175 bytes .../3-digital-features/digital-features.tex | 21 +- .../3-digital-features/digital-features.pdf | Bin 48810 -> 93419 bytes .../3-digital-features/digital-features.tex | 182 +++++++++++++++++- ...arge-numbers-and-central-limit-theorem.pdf | Bin 0 -> 89063 bytes ...arge-numbers-and-central-limit-theorem.tex | 95 ++++++++- .../mathematical-statistics.pdf | Bin 0 -> 90238 bytes .../mathematical-statistics.tex | 168 +++++++++++++++- 10 files changed, 461 insertions(+), 7 deletions(-) create mode 100644 probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.pdf create mode 100644 probability-theory-and-mathematical-statistics/knowledge/4-law-of-large-numbers-and-central-limit-theorem/law-of-large-numbers-and-central-limit-theorem.pdf create mode 100644 probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.pdf 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 d98e415120db4cd094963f58d6511b9e871e47a7..9218ed46b20205a442966960fc92e003e0f7f604 100644 GIT binary patch delta 21502 zcmZU(Q*bV9u&o>0wyhc4wr$(?7u&XNXU4W|+jeHM|9$FQtaZ*sSG`qz*W<0OQT>d= zWw4!Pum%TQVD@H5To@1l@}YOMF>R7@rSF)Bzr4NaES8<&MC@|&g8>9Q>mcmuFhtGe zL+|O_p0u4tOMGj>q_sfB%lpU42gymu*m5B@Sw4slCJH?oHIyBY@a4VAo)J3|2`oP7 z+y;^~2SS-()ic$3K|E7AEcZ*&9F4*EPuB%QTI2>CLTY_bjQBMGmr-V#T`}iW+>5fk z_8^vZ(iT?6;(n*v{{n!UChzi|g6oYZeUIJGy8kto1f&|XUvc*!q67jJI%f5K<#v_h z-AWHXf)q%=U3KUSUoq?#uP|~!Bmre41VwJ42Sltzw?lFhwe$1SNgm$kXOZm$8i{be zg7S!XDEpDF5PA>Dwb?f?e4@=0CMA11R<5}s7qMbv73pGZGo3Yt*!QGlng+6%*o$}I zyn_;3s(YAkZfS^3aiu~gRaSk&+Ud82qs=F^>yYp!wK|hI0!q(_97s$=Sq)PQ$XTrR zhN=83)Pgv|R3ZuCkZL7erL?Al`gfZ3o#JSaW(lML&wCrNjdm6+WAnw(Wb6Y6-`P0e z8*^1w+ND4ME9a2C_>NErHr$b6Du&CheDmBU>LJRPbB=CBv0)(W#VzEOPjy$dfK2>5 zU!V_nY zk&H)a67>#XfflQ*N(Ec!8o1$D-w&iAlfo^tyFFyJ@{OxRaOJu z$Z}QHG-UTUmJHd!YVU8Dh*BRcuR74`OU(m}FveQvI82R_^af^qNolqe)GfFp|EVSi zv9HXaCyjo%B20lt6Jbx6f+EFhDIfpY9Dwi5*uLB=yF}mvnfj)!? zvz9${+dqCOkDDX$>$o2BuXmE!>VV0jxuiIitVCGlsm_Y{LlW%OKF^)nVebB_*VOD&ID z&$41Ygms#=b480%C@r08V0w(UMq$*%?@G7e@#?5k{S*C=u$Csgoa!&PRy<;=wks

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b/probability-theory-and-mathematical-statistics/exercise/2-random-variables-and-distribution/random-variables-and-distribution.tex index 60e0548..d58bd9d 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 @@ -136,7 +136,7 @@ $(X,Y)$联合概率=条件概率×边缘概率。 \draw[black](1,1) -- (1,0) node[below]{$1$}; \draw[black](0,0) -- (1,1); \draw[black, densely dashed](0.5,0.5) -- (0.5,0) node[below]{$\dfrac{1}{2}$}; - \filldraw [fill=gray!50] (0.5,0.5) -- (1,1) -- (1,0) -- (0.5,0.5); + \filldraw [fill=gray!20] (0.5,0.5) -- (1,1) -- (1,0) -- (0.5,0.5); \draw[black](0.8,0.5) node{$D$}; \draw[black, densely dashed](0.6,0) -- (0.6,1) node[above]{$y=y_0$}; \end{tikzpicture} diff --git a/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.pdf 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HcmV?d00001 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 3052c89..70e68f2 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 @@ -23,7 +23,7 @@ \usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} % 超链接 \author{Didnelpsun} -\title{标题} +\title{随机变量数字特征} \date{} \begin{document} \maketitle @@ -34,5 +34,22 @@ \newpage \pagestyle{plain} \setcounter{page}{1} -\section{} +\section{一维随机变量数字特征} + +\subsection{数学期望} + +\subsubsection{离散型随机变量} + +\subsubsection{连续型随机变量} + +\textbf{例题:}连续型随机变量$X$的概率密度为$f(x)=\dfrac{1}{\pi(1+x^2)}$($-\inftyiJldP?C9?L4Tgn@kde^d$O?vshe6EJ#>Lc$LCnU`#Z<)9*xtmHLB`b1+{J>B 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z$2ox&A#QuaD|4ejX0FJo(DSjP#{|Y}N%h2!UItlf(K9xFd&ep#pJ$HK`mnk`U)N<~ zK>pHg1$~3-gz-;(5A$?BdnvG8m4FWiGq4|y7=H~+N22uTS4LnCn6xr5_}>?0TG*|r zD>i-;5f$2sudQM{aL=pcw&c*v>YUsm95UNt3Fkn=_itfKfbnXrOX1LVzy&*G&Xqip zWX%1xZ`j%9-7a(fi?n8Eo*2HJ)aGq23CTDUopsk`m>$sCz{_BlzzF3!~Gogfm_80JI#mCecOlnF)^DjeV#Bq)A7u_;I(iz`+d=jatn6baXEJJbaxvI#1y`Rz+g2AaZU{hkIu#SR8|q z2OG|PJ+gbP>_B1DBQ&0Q3=9Ry#Q%2t2u1Nu$SKsW@}HwgA}p?PpJ{c?F^k4*0_8N9t% z2IobsHi>;H=>(xR?J4Ut@+9E;d$lV>-iYg%gynPSbbID=Ai5_h z$(TWnYGCaHPUH6q7dIdn7@Wo@A~=#Qq0vH5vmz^EYny_1@u9?5d$2kve5i;v{*xJ)@gDV_d;m zHMH#Kl13v>426U;aul9xLRIjI2Kun}ItSwIXEPs867rhGcdAdHq~_;}u6ev}Ebo@! zoh23zA=C>y6J#!B~C zC7`sQ{n1#hV@Z-jr8M=YJT}Z0$o%^FJ>KqKU_Sx=4y{p{?+8n)AgrwfOd~kJcjKv? zz)YLRYU)6&>|sXKxDQf_0yEx$fiFSPo;i`|f`qXZH3omJ(k0xxZ)K%+vb(>oANk6y zb_={X#N>_lO@xv^VIzZ4mS15Cd!tU|nY*)0M(Xw9L~ZMV7#~j>Ry{F{0iN)@n1O&o zl&7IONkTt=$e)mD@;*y(;&J>TTYSni8}U>_vA7ZC%^)@;VbjqxHc7nIocCyUGFNu6 zY%2#d0Sf!%l#gfgSQ9W|mA!{!wnI`HB?N^7^TIu`M6lkIE;;qm`&J> zI9Sb0S=r4DjhQ&OSxt;gjJP@3nZG~r|Jxn9j4Jl_E-=jBlYA9MX*)A}qJKn5-{gfa77D4V+*dS<*Mm@pi%iIZSRisI{;))dd6}^drSnZ@T&1)KdJh8&Bi{iU=%Ds)3EZkEyCf>2LZ+U#!3 z=>lbCuayOf8Hvl}-)jB2Nrz0c_Hd^)BfHmPdaY-q&qvMPg`M?K@;FreropAtY|N}P zqZ8kjTS#8&edGHUbdt`#NM)VgN~~2Y!M%&U$~s6}qH0Y&Kj&H?AsxU*l>%AY<6o1Q k`{d>6jQ`IU?d)Rc0$,有$P\{\vert X-EX\vert\leqslant\epsilon\}\leqslant\dfrac{DX}{\epsilon^2}$或$P\{\vert X-EX\vert<\epsilon\}\geqslant1-\dfrac{DX}{\epsilon^2}$。 + +即一个变量不会离标准差太大距离。 + +可以用于估算随机变量在某范围中取值的概率,也可以证明某些收敛性问题(如数学统计章节中的一致性)。 + +\textbf{例题:}设$XY$为随机变量,数学期望都是2,方差分别为1和4,相关系数为0.5,尝试估计估计概率$P\{\vert X-Y\vert\geqslant6\}$。 + +解:令$Z=X-Y$,$\therefore EZ=E(X-Y)=EX-EY=2-2=0$,所以$P\{\vert X-Y\vert\geqslant6\}=P\{\vert X-Y-0\vert\geqslant6\}=P\{\vert Z-EZ\vert\geqslant6\}\leqslant\dfrac{DZ}{6^2}=\dfrac{3}{36}=\dfrac{1}{2}$。 + +\subsection{常用分布数字特征} + +\begin{center} + \begin{tabular}{|m{50pt}<{\centering}|m{220pt}<{\centering}|c|c|} + \hline + 分布 & 分布列$p_i$或概率密度$f(x)$ & 期望 & 方差 \\ \hline + 0-1分布$B(1,p)$ & $P\{X=k\}=p^k(1-p)^{1-k}$,$k=0,1$ & $p$ & $p(1-p)$ \\ \hline + 二项分布$B(n,p)$ & $P\{X=k\}=C_n^kp^k(1-p)^{n-k}$,$k=0,\cdots,n$ & $np$ & $np(1-p)$ \\ \hline + 泊松分布$P(\lambda)$ & $P\{X=k\}=\dfrac{\lambda^k}{k!}e^{-\lambda}$,$k=0,\cdots$ & $\lambda$ & $\lambda$ \\ \hline + 几何分布$G(p)$ & $P\{X=k\}=(1-p)^{k-1}$,$p,k=1,\cdots$ & $\dfrac{1}{p}$ & $\dfrac{1-p}{p^2}$ \\ \hline + 正态分布$N(\mu,\delta^2)$ & $f(x)=\dfrac{1}{\sqrt{2\pi}\delta}\exp\left\{-\dfrac{(x-\mu)^2}{2\delta^2}\right\}$,$x\in R$ & $\mu$ & $\delta^2$ \\ \hline + 均匀分布$U(a,b)$ & $f(x)=\dfrac{1}{b-a}$,$a0$ & $\dfrac{1}{\lambda}$ & $\dfrac{1}{\lambda^2}$ \\ \hline + \end{tabular} +\end{center} + \section{二维随机变量数字特征} \subsection{数学期望} +\textcolor{violet}{\textbf{定义:}}若$XY$为随机变量,$g(X,Y)$为$XY$的函数,如果$(X,Y)$为离散型随机变量,其联合分布为$p_{ij}=P\{X=x_i,Y=y_i\}$($i,j=1,2,\cdots$),若级数$\sum\limits_i\sum\limits_jg(x_i,y_j)p_{ij}$绝对收敛,则$E[g(X,Y)]=\sum\limits_i\sum\limits_jg(x_i,y_j)p_{ij}$;如果$(X,Y)$为连续型随机变量,其概率密度为$f(x,y)$,若积分$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}g(x,y)f(x,y)\,\textrm{d}x\textrm{d}y$绝对收敛,则定义$E[g(X,Y)]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}g(x,y)f(x,y)\,\textrm{d}x\textrm{d}y$。 + \subsection{协方差相关系数} +\subsubsection{概念} + +\textcolor{violet}{\textbf{定义:}}若随机变量$XY$的方差存在且$DX>0$,$DY>0$,则称$E[(X-EX)(Y-EY)]$为随机变量$X$与$Y$的\textbf{协方差},记为$Cov(X,Y)$,即$Cov(X,Y)=E[(X-EX)(Y-EY)]=E(XY-XEY-YEX+EXEY)=E(XY)-EX\cdot EY$。 + +其中$E(XY)=\left\{\begin{array}{l} + \sum\limits_i\sum\limits_jx_iy_jP\{X=x_i,Y=y_j\} \\ + \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}xyf(x,y)\,\textrm{d}x\textrm{d}y +\end{array}\right.$。 + +从定义来看,方差$DX$就是自己的协方差$Cov(X,X)$。 + +\textcolor{violet}{\textbf{定义:}}$\rho_{XY}=\dfrac{Cov(X,Y)}{\sqrt{DX}\sqrt{DY}}$为随机变量$XY$的\textbf{相关系数}。若$\rho_{XY}=0$,则$XY$不相干,否则相关。 + +相关系数是描述随机变量$XY$之间的线性关系。相关系数为0不代表没有其之间没有关系,也可能存在非线性关系。 + +\subsubsection{性质} + +\begin{itemize} + \item 对称性:$Cov(X,Y)=Cov(Y,X)$,$\rho_{XY}=\rho_{YX}$,$Cov(X,X)=DX$,$\rho_{XX}=1$。 + \item 线性性:$Cov(X,c)=0$,$Cov(aX+b,Y)=aCov(X,Y)$,$Cov(X_1+X_2,Y)=Cov(X_1,Y)+Cov(X_2,Y)$。一般$Cov\left(\sum\limits_{i=1}^na_iX_i,Y\right)=\sum\limits_{i=1}^nCov(X_i,Y)$。 + \item 相关系数有界性:$\vert\rho_{XY}\vert\leqslant1$。 + \item 线性关系下的相关系数:若$Y=aX+b$,则$\rho_{XY}=\left\{\begin{array}{ll} + 1, & a>0 \\ + -1, & a<0 + \end{array}\right.$。 +\end{itemize} + +\textbf{例题:}设随机变量$XY$的概率分布分别为: + +\begin{center} + \begin{tabular}{m{20pt}<{\centering}|m{40pt}<{\centering}m{40pt}<{\centering}} + \hline + $X$ & 0 & 1 \\ \hline + $P$ & 1/3 & 2/3 \\ \hline + \end{tabular}\qquad + \begin{tabular}{m{20pt}<{\centering}|m{40pt}<{\centering}m{40pt}<{\centering}m{40pt}<{\centering}} + \hline + $Y$ & -1 & 0 & 1 \\ \hline + $P$ & 1/3 & 1/3 & 1/3 \\ \hline + \end{tabular} +\end{center} + +且$P\{X^2=Y^2\}=1$。 + +(1)求随机变量$(X,Y)$的概率分布。 + +(2)求$Z=XY$的概率分布。 + +(3)求$XY$的相关系数$\rho_{XY}$。 + +(1)解:根据已知的题目条件可以知道对应的边缘概率分布: + +\begin{center} + \begin{tabular}{c|ccc|c} + \diagbox{$X$}{$Y$} & -1 & 0 & 1 & $X$边缘 \\ \hline + 0 & & & & 1/3 \\ \hline + 1 & & & & 2/3 \\ \hline + $Y$边缘 & 1/3 & 1/3 & 1/3 & 1 \\ \hline + \end{tabular} +\end{center} + +又$P\{X^2=Y^2\}=1$,所以$P\{X^2\neq Y^2\}=0$,所以$X=\pm Y$,解得: + +\begin{center} + \begin{tabular}{c|ccc|c} + \diagbox{$X$}{$Y$} & -1 & 0 & 1 & $X$边缘 \\ \hline + 0 & 0 & 1/3 & 0 & 1/3 \\ \hline + 1 & 1/3 & 0 & 1/3 & 2/3 \\ \hline + $Y$边缘 & 1/3 & 1/3 & 1/3 & 1 \\ \hline + \end{tabular} +\end{center} + +(2)解:$Z=XY$的可能取值为-1,0,1。所以根据表格: + +$P\{Z=-1\}=P\{X=1,Y=-1\}=\dfrac{1}{3}$。 + +$P\{Z=1\}=P\{X=1,Y=1\}=\dfrac{1}{3}$。 + +$P\{Z=0\}=1-P\{Z=1\}-P\{Z=-1\}=\dfrac{1}{3}$。 + +(3)解:$\rho=\dfrac{Cov(X,Y)}{\sqrt{DX}\sqrt{DY}}=\dfrac{EXY-EXEY}{\sqrt{DX}\sqrt{DY}}=0$。 + \section{独立性与相关性} +\begin{itemize} + \item 独立则一定不相关,但是不相关不一定独立。 + \item 如果相关则一定不独立。 + \item 如果$(X,Y)$服从二维正态分布,则$XY$独立与$XY$不相关是充要条件。 +\end{itemize} + \subsection{分布判断独立性} +都是通过分布情况判断独立性: + +\begin{itemize} + \item $F(x,y)=F_X(x)\cdot F_Y(y)$。 + \item $f(x,y)=f_X(x)\cdot f_Y(y)$。 + \item $P\{X=x_i,Y=y_j\}=P\{X=x_i\}\cdot P\{Y=y_j\}$。 +\end{itemize} + \subsection{数字特征判断相关性} +通过相关系数$\rho_{XY}$来判断是否存在线性相关性。 + +$\rho_{XY}=0\Leftrightarrow Cov(X,Y)=0\Leftrightarrow E(XY)=EX\cdot EY\Leftrightarrow D(X\pm Y)=DX+DY$。 + +\subsection{基本判别流程} + +当讨论随机变量$XY$的相关性独立性时: + +\begin{enumerate} + \item 计算$Cov(X,Y)=E(XY)-EXEY$判断是否为0。 + \item 当$Cov(X,Y)\neq0$时则$XY$相关不独立。 + \item 当$Cov(X,Y)=0$时则$XY$不相关。 + \item 若$P(XY)=P(X)P(Y)$则$XY$不相关但独立,否则不相关不独立。 +\end{enumerate} + +\textbf{例题:}设随机变量$X$的概率密度为$f(x)=\dfrac{1}{2}e^{-\vert x\vert}$,$x\in(-\infty,+\infty)$。证明$X$与$\vert X\vert$不相关且不独立。 + +解:$Cov(X,Y)=EXY-EXEY=EX\vert X\vert-EXE\vert X\vert$。 + 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+在满足一定的条件下,大数定律均为$\dfrac{1}{n}\sum\limits_{i=1}^nX_i\overset{P}{\rightarrow}E\left(\dfrac{1}{n}\sum\limits_{i=1}^nX_i\right)$。 + +所以大数定律一般是考定律成立条件与结论正确性。 + +\subsection{切比雪夫大数定律} + +\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,若方差$DX_i$($i\geqslant1$)\textbf{存在且一致有上界},即存在常数$C$,使得$DX_i\leqslant C$对一切$i\geqslant1$均成立,则$\{X_n\}$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$\overline{X}\overset{P}{\rightarrow}E\overline{X}$。 + +\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()。 + +$A.X_1,\dfrac{1}{2}X_2,\cdots,\dfrac{1}{n}X_n,\cdots$\qquad$B.X_1,X_2,\cdots,X_n,\cdots$ + +$C.X_1,2X_2,\cdots,nX_n,\cdots$\qquad$D.X_1,2^2X_2,\cdots,n^2X_n,\cdots$ + +解:切比雪夫大数定律要求有两点,一个是随机变量序列有解,一个是方差存在上界,即$DX_i\leqslant C$。因为题目说明相互独立,所以只用考虑方差上界。 + +$\because X_n\sim E(n)$,$\therefore EX_n=\dfrac{1}{n}$,$DX_n=\dfrac{1}{n^2}$。 + +对于$A$,$D\left(\dfrac{1}{n}X_n\right)=\dfrac{1}{n^2}DX_n=\dfrac{1}{n^4}\leqslant1$。对于$B$,$DX_n=\dfrac{1}{n^2}\leqslant1$。 + +对于$C$,$D(nX_n)=n^2\dfrac{1}{n^2}=1$,对于$D$,$D(n^2X_n)=n^4\dfrac{1}{n^2}=n^2\overset{n\to\infty}{\longrightarrow}\infty$。 + +所以选择$D$。 + +\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$\lim\limits_{x\to\infty}P\left\{\left\vert\dfrac{\mu_n}{n}-p\right\vert<\epsilon\right\}=1$。 + +可以看作通过$n$重伯努利试验,一个事件的概率会逼近一个固定的值。 + +\subsection{辛钦大数定律} + +\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 + +\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 + +$C.$服从同一泊松分布\qquad$D.$服从同一连续型分布 + +解:辛钦大数定律要求三点:随机变量序列独立、拥有同样分布、期望存在。 + +已知题目表示变量相互独立,所以只用证明有同样分布、有期望就可以。 + +对于$BD$而言满足是有分布的,但是此时不一定有期望,所以$BD$不行。 + +对于$A$有相同期望,只要求有期望就可以了,相同期望不一定同一分布。 + +对于$D$服从同一分布,且泊松分布期望存在。 + +\textbf{例题:}将一枚骰子重复投掷$n$次,当$n\to\infty$时,$n$次掷出的点数的算术平均值$\overline{X}$依概率收敛于何值? + +解:根据题目,投掷是独立事件,发生概率是离散的同一分布,且期望存在$=\dfrac{1}{6}\sum\limits_{i=1}^6i=3.5$,所以使用辛钦大数定律。 + +所以根据辛钦大数定律$\overline{X_n}\overset{P}{\rightarrow}E\overline{X_n}=EX_i=3.5$。 + +\section{中心极限定理} + +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$\sum\limits_{i=1}^nX_i\overset{n\to\infty}{\sim}N(n\mu,n\delta^2)$。 + +\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$EX_i=\mu$,$DX_i=\delta^2>0$($i=1,2,\cdots$)存在,则对任意的实数$x$,有$\lim\limits_{n\to\infty}P\left\{\dfrac{\sum\limits_{i=1}^nX_i-n\mu}{\sqrt{n}\delta}\leqslant x\right\}=\dfrac{1}{\sqrt{2}\pi}\int_{-\infty}^xe^{-\frac{t^2}{2}}\,\textrm{d}t=\varPhi(x)$。(正态分布标准化) + +定理要求:独立、同分布、期望方差存在。 + +\subsection{棣莫弗-拉普拉斯定理} + +\textcolor{violet}{\textbf{定义:}}假设随机变量$Y_n\sim B(n,p)$($00.977=\varPhi(2)$。 + +即$\dfrac{5000-50n}{5\sqrt{n}}\geqslant2$,即$n\leqslant98$,即选98。 + \end{document} 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 new file mode 100644 index 0000000000000000000000000000000000000000..796c056d4d4b4c750e399146f444c37e5b001269 GIT binary patch literal 90238 zcma%?Q*1AAM9B)a2d z;lOWym*!Sf(TAd~CE%Q&)+NbW~T2{H`&~b2-xC0Rgi!qHkPxKD_ 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+\definecolor{violet}{RGB}{192,0,255} +\definecolor{aqua}{RGB}{0,255,255} \usepackage{geometry} \setcounter{tocdepth}{4} \setcounter{secnumdepth}{4} @@ -22,6 +25,8 @@ % 数学公式 \usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} % 超链接 +\usepackage{tikz} +% 绘图 \author{Didnelpsun} \title{标题} \date{} @@ -34,5 +39,166 @@ \newpage \pagestyle{plain} \setcounter{page}{1} -\section{} +\section{总体与样本} + +\subsection{总体定义} + +\textcolor{violet}{\textbf{定义:}}研究对象的全体称为\textbf{总体},组成总体的每一个元素称为\textbf{个体}。 + +\subsection{样本} + +\subsubsection{定义} + +\textcolor{violet}{\textbf{定义:}}$n$个相互独立且域总体$X$有相同概率分布的随机变量$X_1,X_2,\cdots,X_n$所组成的整体$(X_1,X_2,\cdots,X_n)$称为来自总体$X$,容量为$n$个一个\textbf{简单随机样本},简称\textbf{样本}。一次抽样结果的$n$个具体值$(x_1,x_2,\cdots,x_n)$称为来自样本$X_1,X_2,\cdots,X_n$的一个\textbf{观测值}或\textbf{样本值}。 + +在概率论中称为独立同分布,而在数理统计就称为简单随机样本。 + +\subsubsection{分布} + +对于容量为$n$的样本$X_1,X_2,\cdots,X_n$有如下定理:假设总体$X$的分布函数为$F(x)$(概率密度为$f(x)$,或概率分布为$p_i=P\{X=x_i\}$),则$(X_1,X_2,\cdots,X_n)$的分布函数为$F(x_1,x_2,\cdots,x_n)=\prod\limits_{i=1}^nF(x_i)$。 + +对于离散型随机变量联合分布:$F(X_1=x_1,X_2=x_2,\cdots,X_n=x_n)=\prod\limits_{i=1}^nP\{X_i=x_i\}$。 + +对于连续型随机变量联合概率密度:$f(x_1,x_2,\cdots,x_n)=\prod\limits_{i=1}^nf(x_i)$。 + +\section{统计量与分布} + +\subsection{统计量} + +设$X_1,X_2,\cdots,X_n$来自总体$X$的一个样本,$g(x_1,x_2,\cdots,x_n)$为$n$元函数,若$g$中不含有任何未知参数,则称$g(X_1,X_2,\cdots,X_n)$为样本$X_1,X_2,\cdots,X_n$的一个\textbf{统计量}。若$(x_1,x_2,\cdots,x_n)$为样本值,则称$g(x_1,x_2,\cdots,x_n)$为$g(X_1,X_2,\cdots,X_n)$的\textbf{观测值}。 + +\subsection{常用统计量} + +\begin{itemize} + \item 样本均值:$\overline{X}=\dfrac{1}{n}\sum\limits_{i=1}^nX_i$。 + \item 样本方差:$S^2=\dfrac{1}{n-1}\sum\limits_{i=1}^n(X_i-\overline{X})^2$。 + \item 样本标准差:$S=\sqrt{\dfrac{1}{n-1}\sum\limits_{i=1}^n(X_i-\overline{X})^2}$。 + \item 样本$k$阶(原点)矩:$A_k=\dfrac{1}{n}\sum\limits_{i=1}^nX_i^k$($k=1,2,\cdots$)。 + \item 样本$k$中心矩:$B_k=\dfrac{1}{n}\sum\limits_{i=1}^n(X_i-\overline{X})^k$($k=1,2,\cdots$)。 +\end{itemize} + +\subsection{顺序统计量} + +\subsubsection{概念} + +将样本$X_1,X_2,\cdots,X_n$的$n$个观测量按其值从小到大的顺序排列,得到$X_{(1)}\leqslant X_{(2)}\leqslant\cdots\leqslant X_{(n)}$。 + +随机变量$X_{(k)}$($k=1,2,\cdots,n$)称为\textbf{第$k$顺序统计量},其中$X_{(1)}$是最小顺序统计量,而$X_{(n)}$是最大顺序统计量。 + +$X_{(n)}$的分布函数为$F_{(n)}(x)=[F(x)]^n$,概率密度为$f_{(n)}(x)=n[F(x)]^{n-1}f(x)$。 + +证明:$F_{(n)}(x)=P\{X_{(n)}\leqslant x\}=P\{\max\{x_1,\cdots,x_n\}\leqslant x\}=P\{x_1\leqslant x,\cdots,x_n\leqslant x\}=P\{x_1\leqslant x\}\cdots P\{x_n\leqslant x\}=F_{(1)}(x)\cdots F_{(n)}(x)=[F(x)]^n$。 + +$X_{(1)}$的分布函数为$F_{(1)}(x)=1-[1-F(x)]^n$,概率密度为$f_{(1)}(x)=n[1-F(x)]^{n-1}f(x)$。 + +证明:$F_{(1)}(x)=P\{X_{(1)}\leqslant x\}=P\{\min\{x_1,\cdots,x_n\}\leqslant x\}=1-P\{\min\{x_1,\cdots$\\$,x_n\}>x\}=1-P\{x_1>x,\cdots,x_n>x\}=1-P\{x_1>x\}\cdots P\{x_n>x\}=1-[1-P\{x_1\leqslant x\}]\cdots[1-P\{x_n\leqslant x\}]=1-[1-F_{(1)}(x)]\cdots[1-F_{(n)}(x)]=1-[1-F(x)]^n$。 + +\subsubsection{性质} + +设总体$X$的期望$EX=\mu$,方差$DX=\delta^2$,样本$X_1,X_2,\cdots,X_n$取自$X$,$\overline{X}$和$S^2$分别为样本的均值和方差,则: + +\begin{itemize} + \item $EX_i=\mu$。 + \item $DX_i=\delta^2$。 + \item $E\overline{X}=EX=\mu$。 + \item $D\overline{X}=D\left(\dfrac{1}{n}\sum\limits_{i=1}^nx_i\right)=\dfrac{1}{n^2}n\delta^2=\dfrac{1}{n}DX=\dfrac{\delta^2}{n}$。 + \item $E(S^2)=E\left(\dfrac{1}{n-1}\sum\limits_{i=1}^n(x_i-\overline{x})^2\right)=E\left(\dfrac{1}{n-1}\sum\limits_{i=1}^n(x_i^2-2x_i\overline{x}+\overline{x}^2)\right)=$\\$E\left(\dfrac{1}{n-1}\left(\sum\limits_{i=1}^nx_i^2-2\overline{x}\cdot\sum\limits_{i=1}^nx_i+n\overline{x}^2\right)\right)=E\left(\dfrac{1}{n-1}\left(\sum\limits_{i=1}^nx_i^2-n\overline{x}^2\right)\right)=$\\$\dfrac{1}{n-1}E\left(\sum\limits_{i=1}^nx_i^2-n\overline{x}^2\right)=\dfrac{1}{n-1}\left(\sum\limits_{i=1}^nEx_i^2-nE\overline{x}^2\right)=\dfrac{n}{n-1}[(Ex_i)^2+Dx_i-(E\overline{x})^2-D\overline{x}]=\dfrac{n}{n-1}\left(\mu^2+\delta^2-\mu^2-\dfrac{\delta^2}{n}\right)=DX=\delta^2$。 +\end{itemize} + +\subsection{三大分布} + +\subsubsection{\texorpdfstring{$\chi^2$分布}{}} + +\paragraph{概念} \leavevmode \medskip + +\textcolor{violet}{\textbf{定义:}}若随机变量$X_1,X_2,\cdots,X_n$相互独立,且都服从标准正态分布,则随机变量$X=\sum\limits_{i=1}^nX_i^2$服从自由度为$n$的$\chi^2$分布,记为$X\sim\chi^2(n)$,特别地$X_i^2\sim\chi^2(1)$。 + +对给定的$\alpha$($0<\alpha<1$)称满足$P\{\chi^2>\chi_\alpha^2(n)\}=\int_{\chi_\alpha^2(n)}^{+\infty}f(x)\,\textrm{d}x=\alpha$的$\chi_\alpha^2(n)$为$\chi^2(n)$分布的\textbf{上$\alpha$分位点}。 + +\begin{tikzpicture}[scale=2] + \draw[-latex](-0.25,0) -- (3,0) node[below]{$x$}; + \draw[-latex](0,-0.25) -- (0,3) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[red, thick, domain=0.125:3] plot (\x,{pow(\x,-0.5)*pow(e,-\x*\x/2)}); + \filldraw[black] (0.5,2.75) node{$n=1$}; + \draw[brown, thick, domain=0.125:3] plot (\x,{pow(e,-\x*\x/2)}); + \filldraw[black] (0.5,1.125) node{$n=2$}; + \draw[blue, thick, domain=0:3] plot (\x,{pow(\x,0.5)*pow(e,-\x*\x/2)}); + \filldraw[black] (0.6,0.45) node{$n=3$}; + \draw[purple, thick, domain=0:3] plot (\x,{pow(\x,4)*pow(e,-\x*\x/2)/5}); + \filldraw[black] (2,0.55) node{$n=10$}; + \filldraw[black] (0.25,3) node{$f(x)$}; +\end{tikzpicture} +\begin{tikzpicture}[scale=2] + \draw[-latex](-0.25,0) -- (3,0) node[below]{$x$}; + \draw[-latex](0,-0.25) -- (0,3) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=0:3] plot (\x,{pow(\x,2)*pow(e,-\x*\x/2)}); + \filldraw[black] (1.7,0.25) node{$\chi_a^2(n)$}; + \filldraw [fill=gray!20] (2,0) -- (2,0.5) -- plot [domain=2:3,smooth] (\x,{pow(\x,2)*pow(e,-\x*\x/2)}) -- (3,0) -- (2,0); + \filldraw[black] (0.25,3) node{$f(x)$}; + \filldraw[black] (2.25,0.2) node{$\alpha$}; +\end{tikzpicture} + +\paragraph{性质} \leavevmode \medskip + +\begin{itemize} + \item 若$X_1\sim\chi^2(n_1)$,$X_2\sim\chi^2(n_2)$,$X_1X_2$相互独立,则$X_1+X_2\sim\chi^2(n_1+n_2)$。一般,若$X_i\sim\chi^2(n_i)$($i=1,2,\cdots,m$),$X_1,X_2,\cdots,X_m$相互独立,则$\sum\limits_{i=1}^mX_i\sim\chi^2\left(\sum\limits_{i=1}^mn_i\right)$。 + \item 若$X\sim\chi^2(n)$,则$EX=n$,$DX=2n$。 +\end{itemize} + +\subsubsection{\texorpdfstring{$t$分布}{}} + +\paragraph{概念} \leavevmode \medskip + +也称为学生分布。 + +若随机变量$X\sim N(0,1)$,$Y\sim\chi^2(n)$,$XY$相互独立,则随机变量$t=\dfrac{X}{\sqrt{Y/n}}$服从自由度为$n$的$t$分布,记为$t\sim t(n)$。 + +当$t\to\infty$时,$t$分布就是正态分布。其是偶函数,所以$Et=0$。 + +\subsubsection{\texorpdfstring{$F$分布}{}} + +若随机变量$X_1,X_2,\cdots,X_n$ + +\subsection{正态总体下结论} + +\section{参数点估计} + +\subsection{概念} + +\subsection{方法} + +\subsubsection{矩估计法} + +\subsubsection{最大似然估计} + +\subsection{估计量平均标准} + +\subsubsection{无偏性} + +\subsubsection{有效性} + +最小方差性。 + +\subsubsection{一致性} + +相合性。 + +\section{参数区间估计与假设检验} + +\subsection{区间估计} + +\subsubsection{概念} + +\subsubsection{正态总体均值的置信空间} + +\subsection{检设检验} + +\subsubsection{思想} + +\subsubsection{正态总体下的六大检验与拒绝域} + +\subsection{两类错误} + \end{document}