概率论与数理统计图解.tex

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egin{document}
    	itle{Huge 概率论与数理统计图解}
    author{dengchaohai}
    maketitle
    
ewpage
    egin{center}
        egin{tikzpicture}
        [r/.style={rectangle,draw,align=left,rounded corners=.8ex}]
        
        
ode(0)at(0,0)[r]{	extbf{0现象}};
        
ode(1)at(5,0)[r]{	extbf{1确定性现象}};
        
ode(2)at(5,-5)[r]{	extbf{2随机性现象}};
        
ode(3)at(10,-5)[r]{	extbf{3随机试验}
            \.可重复
            \.可观察
            \.随机性};
        
ode(4)at(25,-5)[r]{	extbf{4样本点$omega$}};
        
ode(5)at(40,-5)[r]{	extbf{5样本空间$Omega={omega|cdots}$}
            \.离散$Omega={{omega_1,omega_2,cdots}}$
            \.连续$Omega=(a,b)$};
        
ode(6)at(25,-10)[r]{	extbf{6基本事件$omega$}};
        
ode(7)at(40,-10)[r]{	extbf{7事件$A,B,cdots$}
            \$emptysetleq Aleq Omega$};
        
ode(8)at(55,-10)[r]{	extbf{8集合$A,B,cdots$}
            \.互不相容$AB=emptysetRightarrow$对立$overline{A}=Omega-A$
            \.加(交集$Acap B$)减(差集$A-B$)乘(并集$Acup B$)除(包含$Asubseteq B$)
            \.{[(交换律+结合律)=分配律]+自反律}=对偶律};
        
ode(9)at(25,-25)[r]{	extbf{9随机变量$X$}};
        
ode(10)at(40,-25)[r]{	extbf{10概率函数$P(A)$}
            \.$0=emptysetleq P(A)leq Omegaleq 1$
            \.否$P(overline{A})=1-P(A)$
            \.加$P(A+B)=P(A)+P(B)-P(AB)$
            \.减$P(A-B)=P(A)-P(AB)$
            \.乘$P(AB)=P(A)P(B|A)$
            \.除$P(B|A)=frac{P(AB)}{P(A)}$};
        
ode(11)at(55,-25)[r]{	extbf{11分布函数$F(x)$}
            \.单调性$x_1leq x_2Rightarrow F(x_1)leq F(x_2)$
            \.端点极限性$F(-infty)=0,F(+infty)=1$
            \.右连续性$F(x+0)=F(x)$};
        
ode(12)at(15,-25)[r]{	extbf{12随机向量$(X,Y,cdots)$}};
        
ode(13)at(25,-30)[r]{	extbf{13变量函数$Y=g(X)$}};
        
ode(15)at(30,-32.5)[r]{	extbf{15一阶原点矩|期望}
                \.离散$EY=Eg(X)=sum_i^infty g(x_i)p_i$
                \.连续$EY=Eg(X)=int_{-infty}^{+infty} g(x)f(x)dx$
                \.$E(ag(X)+b)=aEg(X)+b$};
        
ode(16)at(30,-35)[r]{	extbf{16二阶中心矩|方差$DY=E(Y-EY)^2=EY^2-(EY)^2$}
            \.$D(aX+b)=a^2DX$};
        
ode(17)at(55,-45)[r]{	extbf{17边缘分布函数$F_X(x)=F(x,+infty),F_Y(y)=F(+infty,y)$}};
        
ode(18)at(55,-35)[r]{	extbf{18联合分布函数$F(x,y)=P{Xleq x,Yleq y}$}};
        
ode(19)at(40,-40)[r]{	extbf{19边缘概率函数}            
            \.离散$p_i^X,p_j^Y$
            \.连续$f_X(x),f_Y(y)$};
        
ode(20)at(40,-35)[r]{	extbf{20联合概率函数}            
            \.离散$p_{ij}=P{X=x_i,Y=y_i}$
            \.连续$f(x,y)$};
        
ode(22)at(32.5,-17.5)[r]{	extbf{22基本概型$P(A)=frac{{omega|omegain A}}{{omega|omegainOmega}}$}
            \.古典概型（有限等可能）
            \.几何概型（无限等可能）};
        
ode(23)at(25,-32.5)[r]{	extbf{23总体$X$}};
        
ode(24)at(18,-32.5)[r]{	extbf{24样本$(X_1,X_2,cdots)$}};
        
ode(25)at(30,-37.5)[r]{	extbf{25切比雪夫不等式$P{|X-EX|geq epsilon}leq frac{DX}{epsilon^2}$}};
        
ode(26)at(47.5,-37.5)[r]{	extbf{26条件概率函数}
            \.离散$P_{i|j}=P{X=x_i|Y=y_j}=frac{P{X=x_i,Y=y_j}}{P{Y=y_j}}=frac{p_{ij}}{P_j^Y}$
            \.连续$f_{X|Y}(x|y)=frac{f(x,y)}{f_Y(y)}$};
        
ode(27)at(60,-37.5)[r]{	extbf{27条件分布函数$F(x|y)=frac{F(x,y)}{F_Y(y)}$}};
        
ode(28)at(40,-32.5)[r]{	extbf{28随机向量的期望,协方差}
            \.离散$EZ=Eg(X,Y)=sum_{i,j}^{infty}g(x_i,y_i)p_{ij}$
            \.连续$EZ=Eg(X,Y)=int_{-infty}^{+infty}int_{-infty}^{+infty}g(x,y)f(x,y)dxdy$
            \.$cov(X,Y)=E[(X-EX)(Y-EY)]$
            \.$E(X+Y)=EX+EY,D(X+Y)=DX+DY+2cov(X,Y)$};
        
ode(29)at(50,-32.5)[r]{	extbf{29条件数学期望}
            \.离散$E[X|Y=y_j]=sum_ix_ip_{i|j}$
            \.连续$E[X|Y=y]=int_{-infty}^{+infty}xf_{X|Y}(x|y)dx$};
        
        draw[->](0)--(1);
        draw[->](0)--(2.5,0)--(2.5,-5)--(2);
        draw[->](2)to node[above]{观察}(3);
        draw[->](3)to node[above]{结果}(4);
        draw[->](4)to node[above]{全体}(5);
        draw[->](4)to node[right]{单个}(6);
        draw[->](5)to node[right]{子集}(7);
        draw[->](6)to node[above]{复合$A={omega|cdots}$}(7);
        draw[->](7)to node[above]{等价}(8);
        draw[->](6)to node[right]{函数$X=X(omega)$}(9);
        draw[->](7)to node[r,right]{测度            
            \.$P(Omega)=1$
            \.$P(A)geq0$
            \.可列可加}(10);
        draw[->](9)to node(21)[r,above]{频率$x=X(omega)Rightarrow P(A)=frac{{omega|omegain A}}{{omega|omegainOmega}}$
            \.离散$p_i=p(x_i)=P{X=x_i}$
            \.连续$f(x)$}(10);
        draw[->](10)to node[r,above]{累和$F(x)=P{Xleq x}$
            \.离散|分段阶梯$sum_i^x p_i$
            \.连续|积分面积$int_{-infty}^x f(x)dx$}(11);
        draw[->](9)to node[right]{复合}(13);
        draw(10)--(40,-30)to node(14)[below]{相乘}(13);
        draw[->](14)to node[right]{累和}(15);
        draw[->](15)--(16);
        draw[->](9)--(12);
        draw[->](12)--(15,-45)--(17);
        draw[->](17)--(18);
        draw[->](16,-25.3)--(16,-40)to node[above]{条件概率$P(B|A)=frac{P(AB)}{P(A)}$|乘法公式$P(AB)=P(A)P(A|B)$|独立性$P(AB)=P(A)P(B)$}(19);
        draw[->](19)to node[right]{全概率|贝叶斯}(20);    
        draw[->](22)--(21);
        draw[->](24.north)to node[above]{假设估计类型,假设估计参数[点估计(最大似然,矩估计),区间估计]}(23.north);
        draw[->](24.south)to node[below,align=left]{无偏(期望)\有效(方差)\相合(依概率收敛$lim_{nlongrightarrowinfty}P{|X_n-X|>epsilon}=0$}(23.south);
        draw[->](23)--(15);
        draw[->](16)--(25);
        draw[->](19)--(26);
        draw[->](26)--(20);    
        draw[->](17)--(27) (27)--(18) (20)--(28) (26)--(29);
        
        end{tikzpicture}
    end{center}    
end{document}