E(X)=2 E(Y)=-2 D(X)=1 D(Y)=4
因为相关系数为-0.5,可得:
E(XY)-(-2)*2 / 1*2 = -0.5
E(XY)=-5 Cov(X,Y)=-1
E(X^2)=D(X)+[E(X)]^2=1+4=5
E(Y^2)=D(Y)+[E(Y)]^2=4+4=8
E[(X+Y)^2]=E(X^2)+E(Y^2)+2E(XY)=5+8-10=3
D(X+Y)=D(X)+D(Y)+2Cox(X,Y)=1+4-2=3
[E(X+Y)]^2=E[(X+Y)^2]-D(X+Y)=0
E(X+Y)=0
下面利用切比雪夫不等式,其中ε=6
P{|X+Y-E(X+Y)|≥6}≤D(X+Y)/36
P{|X+Y-0|≥6}≤3/36
得证:P{|X+Y|≥6}≤1/12
记住以下公式来疯狂推导就可以了~
D(X)=E(X^2)-[E(X)]^2
D(X+Y)=D(X)+D(Y)+2Cov(X,Y)
E[(X+Y)^2]=E(X^2)+E(Y^2)+2E(XY)
相关系数Ρxy=E(XY)-E(X)E(Y) / √D(X)D(Y)
温馨提示:答案为网友推荐,仅供参考