From b761d5528627e0b5e77c3d27f44044961adc046a Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Tue, 13 Jul 2021 23:04:04 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E9=9A=8F=E6=9C=BA=E5=8F=98?= =?UTF-8?q?=E9=87=8F=E4=B8=8E=E5=88=86=E5=B8=83?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../random-variables-and-distribution.pdf | Bin 53219 -> 66872 bytes .../random-variables-and-distribution.tex | 35 +++ .../random-events-and-probability.pdf | Bin 126669 -> 126663 bytes .../random-variables-and-distribution.pdf | Bin 93723 -> 145283 bytes .../random-variables-and-distribution.tex | 263 +++++++++++++++++- 5 files changed, 294 insertions(+), 4 deletions(-) diff --git a/probability-theory-and-mathematical-statistics/exercise/2-random-variables-and-distribution/random-variables-and-distribution.pdf 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zryBPK)tcSbl@%Jgx=p%>stHp=eMGOjz0P~cHeBZq`7EfSGp=tWFG?L=~zMh?IGN~=4v!Z# zRds!`nuzG<*& Z)M{xcke~zz9IRYy>~P<{NhnFe{XY&eh6Dfr diff --git 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 f76265e..b0c36ce 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 @@ -71,4 +71,39 @@ $\therefore p=\left\{X\leqslant\dfrac{1}{2}\right\}=\int_0^\frac{1}{2}2x\,\textr $\therefore P\{Y=k\}=(k-1)\left(\dfrac{1}{8}\right)^2\cdot\left(\dfrac{7}{8}\right)^{k-2}$。 +\section{均匀分布} + +\textbf{例题:}已知随机变量$X\sim U(a,b)$($a>0$)且$P\{04\}=\dfrac{1}{2}$,求$X$的概率密度以及$P\{14\}=\dfrac{1}{2}$,4在其区间中点上,$\dfrac{a+b}{2}=4$。 + +$\because P\{0a\}$。 + +解:$P\{X\leqslant a+1|X>a\}=\dfrac{P\{aa\}}=\dfrac{\int_a^{a+1}e^{-x}\,\textrm{d}x}{\int_a^{+\infty}e^{-x}\,\textrm{d}x}=1-\dfrac{1}{e}$。 + +也可以根据指数分布的无记忆性:$P\{X\leqslant a+1|X>a\}=1-P\{X>a+1|X>a\}=1-P\{X>1\}=P\{X\leqslant1\}=F(1)=1-\dfrac{1}{e}$。 + +\section{正态分布} + +\textbf{例题:}已知随机变量$X\sim N(0,1)$,对给定的$\alpha$($0<\alpha>1$),数$\mu_\alpha$满足$P\{X>\mu_\alpha\}=\alpha$,若$P\{\vert X\vert\mu_\alpha\}=\alpha$即表示$\mu_\alpha$为标准正态分布的上$\alpha$分位点。 + +又$P\{\vert X\vertx\}=\dfrac{1-\alpha}{2}$。 + +$\because$面积为$\alpha$的下标为$\alpha$,$\therefore$面积为$\dfrac{1-\alpha}{2}$的下标为$\dfrac{1-\alpha}{2}$,$x=\mu_\frac{1-\alpha}{2}$。 + \end{document} diff --git a/probability-theory-and-mathematical-statistics/knowledge/1-random-events-and-probability/random-events-and-probability.pdf 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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 @@ -1,6 +1,6 @@ \documentclass[UTF8, 12pt]{ctexart} % UTF8编码,ctexart现实中文 -\usepackage{color} +\usepackage{xcolor} % 使用颜色 \definecolor{orange}{RGB}{255,127,0} \definecolor{violet}{RGB}{192,0,255} @@ -69,7 +69,7 @@ \item $P(X=a)=F(a)-F(a-0)$。$\because P\{X\leqslant a\}=P\{X0 \\ + 0, & \text{其他} +\end{array}\right.$,$F(x)=\left\{\begin{array}{ll} + 1-e^{-\lambda x}, & x\geqslant0 \\ + 0, & x<0 \\ +\end{array}\right.$,则称$X$在区间$(a,b)$上服从参数为$\lambda$的\textbf{指数分布},记为$X\sim E(\lambda)$。 + +\begin{tikzpicture}[scale=2] + \draw[-latex](-0.25,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-0.25) -- (0,1.25) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=0:2] plot (\x,{pow(e,-\x)}); + \filldraw[black] (1,1) node{$f(x)$}; +\end{tikzpicture} +\hspace{2.5em} +\begin{tikzpicture}[scale=2] + \draw[-latex](-0.25,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-0.25) -- (0,1.25) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, densely dashed](2,1) -- (0,1) node[left]{$1$}; + \draw[black, thick, domain=0:2] plot (\x,{1-pow(e,-\x)}); + \filldraw[black] (0.25,1.25) node{$F(x)$}; +\end{tikzpicture} + +指数分布中$\lambda$代表失效率,往往用来代表一个事物毁坏的过程,如灯泡毁坏。 + +\textcolor{aqua}{\textbf{定理:}}无记忆性:若$X$服从指数分布,则$P\{X>s+t|X>s\}=P\{X>t\}$。 + +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。 + +证明:$P\{X>s+t|X>s\}=\dfrac{P\{X>s+t\}}{P\{X>s\}}=\dfrac{1-P\{X\leqslant s+t\}}{1-P\{X\leqslant s\}}$ + +$=\dfrac{1-F(s+t)}{1-F(x)}=\dfrac{1-(1-e^{-\lambda(s+t)})}{1-(1-e^{-\lambda s})}=\dfrac{e^{-\lambda(s+t)}}{e^{-\lambda s}}=e^{-\lambda t}=1-(1-e^{-\lambda t})$ + +$=1-F(t)=1-P\{X\leqslant t\}=P\{X>t\}$。 + \subsubsection{正态分布} +\textcolor{violet}{\textbf{定义:}}如果$X$的概率密度为$f(x)=\dfrac{1}{\sqrt{2\pi\delta}}e^{-\frac{1}{2}(\frac{x-\mu}{\delta})^2}$($-\infty0$),则称$X$服从参数为$(\mu,\delta^2)$的\textbf{正态分布},称$X$为\textbf{正态变量},记为$X\sim N(\mu,\delta^2)$。 + +$f(x)$的图形关于$x=\mu$对称,即$f(\mu-x)=f(\mu+x)$,并在$x=\mu$处有唯一最大值$f(\mu)=\dfrac{1}{\sqrt{2\pi}\delta}$。$\mu-\delta$和$\mu+\delta$为拐点。 + +\begin{tikzpicture}[scale=2] + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-0.25) -- (0,1.25) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=-2:2] plot (\x,{pow(e,-\x*\x/2)}); + \draw[black, densely dashed](-1,0.65) -- (-1,0) node[below]{$\mu-\delta$}; + \draw[black, densely dashed](1,0.65) -- (1,0) node[below]{$\mu+\delta$}; + \draw[black, densely dashed](0,1) -- (-1,1) node[left]{$\dfrac{1}{\sqrt{2\pi}\delta}$}; + \filldraw[black] (0.35,0.25) node{$x=\mu$}; + \filldraw[black] (1,1) node{$f(x)$}; +\end{tikzpicture} + +当$\mu=0$,$\delta=1$时的正态分布$N(0,1)=\dfrac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$为\textbf{标准正态分布},记为$\phi(x)$,分布函数为$\varPhi(x)=\displaystyle{\int_{-\infty}^x\dfrac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}\,\textrm{d}t}$。$\phi(x)$为偶函数,$\varPhi(0)=\dfrac{1}{2}$,$\varPhi(-x)=1-\varPhi(x)$。 + +若$X\sim N(0,1)$,$P\{X>\mu_\alpha\}=\alpha$,则称$\mu_\alpha$为标准正态分布的\textbf{上侧$\alpha$分位数/上$\alpha$分位点}。 + +若$X\sim N(\mu,\delta^2)$,则 + +\begin{itemize} + \item $F(x)=P\{X\leqslant x\}=P\left\{\dfrac{X-\mu}{\delta}\leqslant\dfrac{x-\mu}{\delta}\right\}=\varPhi\left(\dfrac{x-\mu}{\delta}\right)$。(标准化) + \item $F(\mu-x)+F(\mu+x)=1$。 + \item $P\{a1$恒成立,所以$X^2+1\leqslant y$不可能发生,概率为0,所以$F_Y(y)=0$。 + +当$y>2$时,$Y=X^2+1$在$X\in[-1,1]$时$Y\in[1,2]$,所以$X^2+1\leqslant y$必然成立,所以所以$F_Y(y)=1$。 + +当$10$,则称$P\{X=x_i|Y=y_j\}=\dfrac{P\{X=x_i,Y=y_j\}}{P\{Y=y_j\}}=\dfrac{p_{ij}}{p_{\cdot j}}$($i=1,2,\cdots$)为$X$在$Y=y_j$条件下的\textbf{条件分布}。 + +同理\textcolor{violet}{\textbf{定义:}}$Y$在$X=x_i$条件下的\textbf{条件分布}为$P\{Y=y_j|X=x_i\}=\dfrac{p_{ij}}{p_{i\cdot}}$($j=1,2,\cdots$)。 + +\section{二维连续型随机变量} + +\textcolor{violet}{\textbf{定义:}}如果二维随机变量$(X,Y)$的联合分布函数$F(x,y)$可表示为$F(x,y)=\int_{-\infty}^x\int_{-\infty}^yf(u,v)\,\textrm{d}u\textrm{d}v$,($(x,y)\int R^2$),其中$f(x,y)$为非负可积函数,则称$(X,Y)$为\textbf{二维连续型随机变量},$f(x,y)$为$(X,Y)$的\textbf{概率密度},记为$(X,Y)\sim f(x,y)$。 + +二元函数$f(x,y)$是概率密度的充要条件$f(x,y)\geqslant0$,$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}f(x,y)\,\textrm{d}x\textrm{d}y$\\$=1$。 + +改变$f(x,y)$有限个点值(仍取非负值),$f(x,y)$仍是概率密度。 + +\subsection{联合概率密度} + +\textcolor{violet}{\textbf{定义:}}设$(X,Y)$的联合分布函数为$F(x,y)$,概率密度为$f(x,y)$,则 + +\begin{itemize} + \item $F(x,y)$为$(x,y)$的二元连续函数,且$F(x,y)=P\{X\leqslant x,Y\leqslant y\}=\int_{-\infty}^x\int_{-\infty}^xf(u,v)\,\textrm{d}u\textrm{d}v$。 + \item 设$G$为平面上某个区域,则$P\{(X,Y)\in G\}=\iint\limits_Gf(x,y)\,\textrm{d}x\textrm{d}y$。 + \item 若$f(x,y)$在点$(x,y)$处连续,则$\dfrac{\partial^2F(x,y)}{\partial x\partial y}=f(x,y)$。 + \item 若$F(x,y)$连续可导,则$(X,Y)$是连续型随机变量,则$\dfrac{\partial^2F(x,y)}{\partial x\partial y}$是其概率密度。 +\end{itemize} + +\subsection{边缘概率密度} + +\textcolor{violet}{\textbf{定义:}}设$(X,Y)\sim f(x,y)$,则$X$的边缘分布函数为$F_X(x)=F(x,+\infty)=\int_{-\infty}^x\left[\int_{-\infty}^{+\infty}f(u,v)\,\textrm{d}v\right]\textrm{d}u$,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$也为连续型随机变量,其概率密度为$f_Y(y)=\int_{-\infty}^{+\infty}f(x,y)\,\textrm{d}x$。 + +\subsection{条件概率密度} + +\textcolor{violet}{\textbf{定义:}}设$(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)$。 + +\textcolor{violet}{\textbf{定义:}}$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=\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)$ + +\subsection{二维均匀分布} + +\subsection{二维正态分布} + +\section{随机变量独立性} + +\subsection{概念} + +\subsection{充要条件} + +\subsection{性质} + +\section{二维随机变量函数分布} + +\subsection{离散型} + +\subsection{连续型} + +\subsection{混合型} + \end{document}