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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 45fa3a7..322251e 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 @@ -222,7 +222,118 @@ $\because$面积为$\alpha$的下标为$\alpha$,$\therefore$面积为$\dfrac{1 使用定义法则直接用二重积分的分布函数来求,使用卷积公式则使用概率密度。 -\subsection{二维随机变量分布} +\subsection{二维离散型随机变量} + +\subsection{二维连续型随机变量} + +\subsubsection{联合概率} + +\paragraph{概率函数} \leavevmode \medskip + +已知联合概率密度,可以求概率函数,通过二重积分的方式,图像面积即是概率。 + +\textbf{例题:}已知概率密度为$f(x,y)=\left\{\begin{array}{ll} + 6, & 0\dfrac{1}{2}\right\}$,$P\left\{Y<\dfrac{1}{2}\right\}$。 + +解: + +\begin{minipage}{0.45\linewidth} + \begin{tikzpicture}[scale=3] + \begin{scope} + \clip(0.5,0)rectangle(1,1); + \draw[black, thick, smooth, domain=0:1,fill=gray!20] plot (\x, {\x*\x}); + \end{scope} + \draw[black, thick, smooth, domain=0:1] plot (\x, {\x*\x}); + \draw[black, thick, smooth, domain=0:1] plot (\x, {\x}); + \draw[black, thick](1/2,1) -- (1/2,0) node[below]{$\frac{1}{2}$}; + \draw[-latex](0,0) -- (1.25,0) node[below]{$x$}; + \draw[-latex](0,-0.25) -- (0,1.25) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick](1,1) -- (0,1) node[left]{$1$}; + \draw[black, thick](1,1) -- (1,0) node[below]{$1$}; + \filldraw[black] (0.25,0.75) node{$y=x$}; + \filldraw[black] (0.75,0.25) node{$y=x^2$}; + \end{tikzpicture} +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{tikzpicture}[scale=3] + \begin{scope} + \clip(0,0)rectangle(1,0.5); + \draw[black, thick, smooth, domain=0:1,fill=gray!20] plot (\x, {\x*\x}); + \end{scope} + \draw[black, thick, smooth, domain=0:1] plot (\x, {\x*\x}); + \draw[black, thick, smooth, domain=0:1] plot (\x, {\x}); + \draw[black, thick](1,1/2) -- (0,1/2) node[left]{$\frac{1}{2}$}; + \draw[-latex](0,0) -- (1.25,0) node[below]{$x$}; + \draw[-latex](0,-0.25) -- (0,1.25) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick](1,1) -- (0,1) node[left]{$1$}; + \draw[black, thick](1,1) -- (1,0) node[below]{$1$}; + \filldraw[black] (0.25,0.75) node{$y=x$}; + \filldraw[black] (0.75,0.25) node{$y=x^2$}; + \end{tikzpicture} +\end{minipage} + +$P\left\{X>\dfrac{1}{2}\right\}=\displaystyle{6\int_{\frac{1}{2}}^1\textrm{d}x\int_{x^2}^x\textrm{d}y=6\int_{\frac{1}{2}}^1(x-x^2)\textrm{d}x=6\left(\dfrac{1}{2}x^2-\dfrac{1}{3}x^3\right)\bigg\vert_{\frac{1}{2}}^1=\dfrac{1}{2}}$。 + +$P\left\{Y<\dfrac{1}{2}\right\}=\displaystyle{6\int^{\frac{1}{2}}_0\textrm{d}y\int_y^{\sqrt{y}}\textrm{d}x=6\int^{\frac{1}{2}}_0(\sqrt{y}-y)\textrm{d}y=\sqrt{2}-\dfrac{3}{4}}$。 + +\subsubsection{边缘概率} + +\paragraph{边缘概率函数} \leavevmode \medskip + +往往是已知联合概率函数$F(x,y)$求边缘概率函数$F_X(x)$、$F_Y(y)$,需要将联合概率函数中的$x/y\to+\infty$,然后求这个函数的极限值。 + +\textbf{例题:}如果二维随机变量$(X,Y)$的分布函数为$F(x,y)=\\\left\{\begin{array}{ll} + 1-e^{-\lambda_1x}-e^{-\lambda_2y}+e^{-\lambda_1x-\lambda_2y-\lambda12\max\{x,y\}}, & \lambda_1,\lambda_2,\lambda_{12}>0,x>0,y>0 \\ + 0, & \text{其他} +\end{array}\right.$,\\求$XY$各自的边缘分布函数。 + +解:$\lim\limits_{x\to+\infty}\max\{x,y\}=x=+\infty$,$\lim\limits_{y\to+\infty}\max\{x,y\}=y=+\infty$。 + +$F_X(x)=F(x,+\infty)=1-e^{-\lambda_1x}-0+0=1-e^{-\lambda_1x}$,$x>0$,当其他时$=0$。 + +$F_Y(y)=F(+\infty,y)=1-0-e^{-\lambda_2y}+0=1-e^{-\lambda_2x}$,$y>0$,当其他时$=0$。 + +\paragraph{边缘概率密度} \leavevmode \medskip + +往往是已知联合概率密度$f(x,y)$求边缘概率密度$f_X(x)$、$f_Y(y)$,需要将联合概率密度对另一个变量进行上下限无穷的一重积分,如果$xy$有上下限的定义域则需要画出图像取交集。 + +确定上下限时要注意,如果求$x$的边缘分布对$y$积分,表示$x$不动,求$y$的范围,求$y$的则反之。 + +\textbf{例题:}求$f(x,y)=\left\{\begin{array}{ll} + \dfrac{5}{4}(x^2+y), & 0