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b/probability-theory-and-mathematical-statistics/exercise/2-random-variables-and-distribution/random-variables-and-distribution.tex @@ -39,7 +39,13 @@ \pagestyle{plain} \setcounter{page}{1} -\section{二项分布} +分布函数变量区域左闭右开,概率密度则不要求。 + +\section{一维随机变量} + +\subsection{一维随机变量分布} + +\subsubsection{二项分布} $P\{X=k\}=C_n^kp^k(1-p)^{n-k}$($k=0,1,\cdots,n$,$00$),$X\sim P(\lambda)$。 @@ -64,7 +70,7 @@ $P\{X=k\}=\dfrac{\lambda^k}{k!}e^{-\lambda}$($k=0,1,\cdots,n$,$\lambda>0$) 由于随机抽四页类似于伯努利试验是相互独立的,所以随机抽4页都无错误的概率为$[P\{X=0\}]^4=e^{-8}$。 -\section{几何分布} +\subsubsection{几何分布} $P\{X=k\}=(1-p)^{k-1}p$($k=0,1,\cdots,n$,$00 \\ @@ -190,9 +196,9 @@ $f(x)=\left\{\begin{array}{ll} 则$P\{3>X>2|X>1\}=\dfrac{P\{3>X>2\}}{P\{X>1\}}=\dfrac{P\{X>2\}-P\{X>3\}}{P\{X>1\}}=\dfrac{e^{-2}-e^{-3}}{e^{-1}}=e^{-1}-e^{-2}$。 -\section{正态分布} +\subsubsection{正态分布} -$f(x)=\dfrac{1}{\sqrt{2\pi\sigma}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$($-\infty0$),$X\sim N(\mu,\sigma^2)$。 +$f(x)=\dfrac{1}{\sqrt{2\pi}\sigma}\exp\left(-\dfrac{(x-\mu)^2}{2\sigma^2}\right)$($-\infty0$),$X\sim N(\mu,\sigma^2)$。 \textbf{例题:}已知随机变量$X\sim N(0,1)$,对给定的$\alpha$($0<\alpha>1$),数$\mu_\alpha$满足$P\{X>\mu_\alpha\}=\alpha$,若$P\{\vert X\vert0$,$-1<\rho<1$。 + +\paragraph{正态分布性质} \leavevmode \medskip + +\textbf{例题:}$(X,Y)\sim N(\mu_1,\mu_2;\sigma_1^2,\sigma_2^2;0)$,分布函数为$F(x,y)$,已知$F(\mu_1,y)=\dfrac{1}{4}$,求$y$。 + +解:当$\rho=0$时,$F(X,Y)=\dfrac{1}{2\pi\sigma_1\sigma_2}\exp\left(-\dfrac{1}{2}\left(\left(\dfrac{x-\mu_1}{\sigma_1}\right)^2+\left(\dfrac{y-\mu_2}{\sigma_2}\right)^2\right)\right)\\=F_X(x)F_Y(y)$,即$XY$相互独立。 + +$X\sim N(\mu_1,\sigma_1)$,$Y\sim N(\mu_2,\sigma_2)$,$F_X(\mu_1)=P\{X\leqslant\mu_1\}=\dfrac{1}{2}$。 + +$F(\mu_1,y)=F_X(\mu_1)F_Y(y)=\dfrac{1}{2}F_Y(y)=\dfrac{1}{4}$,则$F_Y(y)=\dfrac{1}{2}$,即根据性质$y=\mu_2$。 + +\paragraph{标准正态化} \leavevmode \medskip + +$F(x)=P\{X\leqslant x\}=P\left\{\dfrac{X-\mu}{\sigma}\leqslant\dfrac{x-\mu}{\sigma}\right\}=\varPhi\left(\dfrac{x-\mu}{\sigma}\right)$。 + +即将$XY$的相关系数消去。 + +\textbf{例题:}设随机变量$(X,Y)$的分布函数为$\varPhi(2x+1)\cdot\varPhi(2y-1)$,其中$\varPhi(x)$为标准正态分布函数,求$(X,Y)$的分布函数。 + +解:由分布函数为$\varPhi(2x+1)\varPhi(2y-1)$是$X$的分布函数和$Y$的分布函数的乘积,所以可知$XY$相互独立。 + +所以根据标准化公式:$\varPhi(2x+1)\cdot\varPhi(2y-1)=\varPhi\left(\dfrac{x+\dfrac{1}{2}}{\dfrac{1}{2}}\right)\varPhi\left(\dfrac{y-\dfrac{1}{2}}{\dfrac{1}{2}}\right)$。 + +$\therefore(X,Y)\sim N\left(-\dfrac{1}{2},\dfrac{1}{2};\dfrac{1}{4},\dfrac{1}{4};0\right)$。 + +\subsection{二维随机变量函数分布} + +\subsubsection{和的分布} + +\textbf{例题:}随机变量$(X,Y)$的概率密度函数$f(x,y)=\left\{\begin{array}{ll} + e^{-y}, & 0i<_G|n3*%n zncG{sT9NRwatR5+xwtx;8{5Hou3u(N#%-`5NIZYSeT%3ikv)P$X=7MHpVW%DOJMsC z>ATH+@1*2O2mg#TolNEs8O6bIN0Uu|5-g+fsuIT3CMDbb=NTf+S!0Y!aa&7rYErA^mN)3_uv|Dk5V)v8z#AM`Xb^R zD4LvCD^1zauX@eC*DutY>m-fiLh%=6a<`Pr*Rbzu>#py;ovy01IihjqwURpUy!kD5 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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 index 0439155..f6d5fc2 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 @@ -275,7 +275,7 @@ $=1-F(t)=1-P\{X\leqslant t\}=P\{X>t\}$。 \subsubsection{正态分布} -\textcolor{violet}{\textbf{定义:}}如果$X$的概率密度为$f(x)=\dfrac{1}{\sqrt{2\pi\sigma}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$($-\infty0$),则称$X$服从参数为$(\mu,\sigma^2)$的\textbf{正态分布},称$X$为\textbf{正态变量},记为$X\sim N(\mu,\sigma^2)$。 +\textcolor{violet}{\textbf{定义:}}如果$X$的概率密度为$f(x)=\dfrac{1}{\sqrt{2\pi}\sigma}\exp\left(-\dfrac{(x-\mu)^2}{2\sigma^2}\right)$($-\infty0$),则称$X$服从参数为$(\mu,\sigma^2)$的\textbf{正态分布},称$X$为\textbf{正态变量},记为$X\sim N(\mu,\sigma^2)$。 $f(x)$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$f(\mu)=\dfrac{1}{\sqrt{2\pi}\sigma}$。$\mu-\sigma$和$\mu+\sigma$为拐点。 @@ -291,7 +291,7 @@ $f(x)$的图形关于$x=\mu$对称,即$f(\mu-x)=f(\mu+x)$,并在$x=\mu$处 \filldraw[black] (1,1) node{$f(x)$}; \end{tikzpicture} -当$\mu=0$,$\sigma=1$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,$\varPhi(0)=\dfrac{1}{2}$,$\varPhi(-x)=1-\varPhi(x)$。 +当$\mu=0$,$\sigma=1$时的正态分布$N(0,1)=\dfrac{1}{\sqrt{2\pi}}\exp\left(-\dfrac{x^2}{2}\right)$为\textbf{标准正态分布},记为$\varPhi(x)$,$\varPhi(x)$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}。 @@ -561,6 +561,16 @@ $p_{\cdot j}=P\{Y=y_i\}=\sum\limits_{i=1}^\infty P\{X=x_i,Y=y_j\}=\sum\limits_{i 若$X_1,X_2,\cdots,X_n$相互独立,则其各自的函数$g_1(X_1)g_2(X_2)\cdots g_n(X_n)$也相互独立。 +\subsection{独立同分布运算} + +若$XY$相互独立,则: + +\begin{enumerate} + \item 若$X\sim B(n,p)$,$Y\sim B(m,p)$,则$X+Y\sim B(n+m,p)$。 + \item 若$X\sim P(\lambda_1)$,$Y\sim P(\lambda_2)$,则$X+Y\sim P(\lambda_1+\lambda_2)$。 + \item 若$X\sim N(\mu_1,\sigma_1^2)$,$Y\sim N(\mu_2,\sigma_2^2)$,则$X+Y\sim N(\mu_1+\mu_2,\sigma_1^2+\sigma_2^2)$,$X-Y\sim N(\mu_1-\mu_2,\sigma_1^2+\sigma_2^2)$。 +\end{enumerate} + \section{二维随机变量函数分布} \textcolor{violet}{\textbf{定义:}}设$X,Y$为随机变量,$g(x,y)$为二元函数,则以随机变量$X,Y$作为变量的函数$U=g(X,Y)$也是随机变量,称为\textbf{随机变量$X,Y$的函数}。 @@ -615,13 +625,15 @@ $Y_2$的概率分布为$Y_2\sim\left(\begin{array}{ccccc} 对于(连续型,连续型)随机变量函数分布也是连续型。可以采用分布函数法和卷积公式。 +设$(X,Y)$的概率密度为$f(x,y)$,$Z=g(x,y)$,则$Z$的分布函数为$F_Z(z)=P(Z\leqslant z)=P(g(X,Y)\leqslant z)=\iint\limits_{g(x,y)\leqslant z}f(x,y)\,\textrm{d}x\textrm{d}y$,若$Z$仍然为联系型随机变量,则$Z$的概率密度为$f_Z(z)=F_Z'(z)$。 + \subsubsection{和的分布} \textcolor{aqua}{\textbf{定理:}}设$(X,Y)\sim f(x,y)$,则$Z=X+Y$的概率密度为$f_Z(z)=\int_{-\infty}^{+\infty}f(x,z-x)\,\textrm{d}x=\int_{-\infty}^{+\infty}f(z-y,y)\,\textrm{d}y$。 证明:利用定义法$F_Z(z)=P\{Z\leqslant z\}=P\{X+Y\leqslant z\}=\iint\limits_Df(x,y)\,\textrm{d}x\textrm{d}y$,$D=x+y\leqslant z$。 -$=\int_{-\infty}^{+\infty}\textrm{d}x\int_{-\infty}^{z-x}f(x,y)\,\textrm{d}y$。又$f_Z(z)=F_Z'(z)$,$=(\int_{-\infty}^{+\infty}\textrm{d}x\int_{-\infty}^{z-x}f(x,y)\,\textrm{d}y)'_z$ +$=\int_{-\infty}^{+\infty}\textrm{d}x\int_{-\infty}^{z-x}f(x,y)\,\textrm{d}y$。又$f_Z(z)=F_Z'(z)$,$=(\int_{-\infty}^{+\infty}\textrm{d}x\int_{-\infty}^{z-x}f(x,y)\,\textrm{d}y)'_z$。 $=\int_{-\infty}^{+\infty}(\int_{-\infty}^{z-x}f(x,y)\,\textrm{d}y)'_z\textrm{d}x=\int_{-\infty}^{+\infty}f(x,z-x)\,\textrm{d}x$ @@ -639,6 +651,12 @@ $=\int_{-\infty}^{+\infty}(\int_{-\infty}^{z-x}f(x,y)\,\textrm{d}y)'_z\textrm{d} 这里不积$y$只积$x$,因为$x=zy$,所以积分简单从而只积$x$。 +\subsubsection{最值的分布} + +需要根据最值的定义得到$XY$的概率分布。设$(X,Y)$的联合分布函数为$F(x,y)$,$XY$的分布函数分别为$F_X(x)$和$F_Y(y)$,则$Z=\max(X,Y)$的分布函数为$F_{max}(z)=P\{\max(X,Y)\leqslant z\}=P\{X\leqslant z,Y\leqslant z\}=F(z,z)$,$Z=\min(X,Y)$的分布函数为$F_{min}(z)=P\{\min(X,Y)\leqslant z\}=P\{(X\leqslant z)\cup(Y\leqslant z)\}=P\{X\leqslant z\}+P\{Y\leqslant z\}-P\{X\leqslant z,Y\leqslant z\}=F_X(z)+F_Y(z)-F(z,z)$。 + +如$P\{\min(X,Y)\geqslant 1\}=P\{X\geqslant1,Y\geqslant1\}=P\{X\geqslant1\}P\{Y\geqslant1\}$。 + \subsubsection{卷积公式} 积谁不换谁,换完求偏导。 @@ -734,6 +752,8 @@ $f_X(x)f_Y(z-x)=\left\{\begin{array}{ll} 即使用全集分解思想解决,若$A$是离散型随机变量,$B$是连续型随机变量,则$P(B)=P(B\Omega)=P(BA_1\cup BA_2\cup\cdots)=P(BA_1)+P(BA_2)\cdots=P(A_1)P(B|A_1)+P(A_2)P(B|A_2)+\cdots$。这就是全概率公式。 +设$X$的分布律为$P\{X=x_i\}=P_i$,$Y$为连续型随机变量,$Z=g(X,Y)$,则$Z$的分布函数为$F_Z(z)=P\{Z\leqslant z\}=P\{g(X,Y)\leqslant z\}=\sum\limits_iP\{X=x_i\}P\{g(X,Y)\leqslant z|X=x_i\}=\sum\limits_iP_iP\{g(x_i,Y)\leqslant z\}$。 + \textbf{例题:}设随机变量$X$与$Y$相互独立,其中$X$概率分布为$\left(\begin{array}{cc} 1 & 2 \\ 0.3 & 0.7