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1.期望

定义

\[e(x)=\ sum \ limits _ { k=1 } { \ infty } x _ KP _ k-离散类型

\]

\[e(x)=\ int _ {-\ infty } xf(x)dx-连续类型

\]

性质

\ (e (c)=c,c是常数\)

\ (e (CX)=ce (x),c是常数\)

\(E(X Y)=E(X) E(Y)\)

\(如果X和Y相互独立,则有E(XY)=E(X)E(Y)\)

2.方差

定义

\[d(x)=\ sum \ limits _ { k=1 } { \ infty }[x _ k-e(x)]2p _ k-离散型

\]

\[d(x)=\int_{-\infty}^{ \infty}[x-e(x)]^2f(x)dx

\]

性质

\(D(X)=E(X^2)-[E(X)]^2\)

\ (d (c)=0,c是常数\)

\ (d (CX)=c 2d (x),c是常数\)

\ (d (x c)=d (x),c是常数\)

\(两个随机变量有\)

\[D(X Y)=D(X)D(Y)2E[(X-E(X)(Y-E(Y))]

\]

\(如果两个随机变量是独立的,是\)

\[D(X Y)=D(X) D(Y)

\]

3.协方差

定义

\(两个随机变量X,Y的协方差\)

\[cov(X,Y)=E[(X-E(X)(Y-E(Y))]

\]

\(相关系数\)

\[\rho_{XY}=\frac{cov(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}}

\]

性质

\(cov(aX,bY)=ab\ cov(X,Y),a,b为常数\)

\(cov(X_1 X_2,Y)=cov(X_1,Y) cov(X_2,Y)\)

\(独立一定无关,无关不一定独立\)

协方差矩阵

\(它必须是对称和半正的\)

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