题目

设随机变量XY的联合分布律为X IY 0 1 2-|||-0 0.2 0.1 0.1-|||-1 0.1 0 0.1-|||-2 0 0.3 0.1求X IY 0 1 2-|||-0 0.2 0.1 0.1-|||-1 0.1 0 0.1-|||-2 0 0.3 0.1.

设随机变量XY的联合分布律为

求 $E\left(X\right),E\left(Y\right),E\left(X-Y\right),D\left(X\right),D\left(Y\right),Cov\left(X,Y\right)$ .

题目解答

答案

X的边缘分布律为 $P\left(X=0\right)=P\left(X=0,Y=0\right)+P\left(X=0,Y=1\right)+P\left(X=0,Y=2\right)$ $=0.2+0.1+0.1=0.4$ ， $P\left(X=1\right)=P\left(X=1,Y=0\right)+P\left(X=1,Y=1\right)+P\left(X=1,Y=2\right)$ $=0.1+0+0.1=0.2$ ， $P\left(X=2\right)=P\left(X=2,Y=0\right)+P\left(X=2,Y=1\right)+P\left(X=2,Y=2\right)$ $=0+0.3+0.1=0.4$ ，则 $E\left(X\right)=0\times P\left(X=0\right)+1\times P\left(X=1\right)+2\times P\left(X=2\right)$ $=0\times0.4+1\times0.2+2\times0.4=1$ ，

Y的边缘分布律为 $P\left(Y=0\right)=P\left(X=0,Y=0\right)+P\left(X=1,Y=0\right)+P\left(X=2,Y=0\right)$ $=0.2+0.1+0=0.3$ ， $P\left(Y=1\right)=P\left(X=0,Y=1\right)+P\left(X=1,Y=1\right)+P\left(X=2,Y=1\right)$ $=0.1+0+0.3=0.4$ ， $P\left(Y=2\right)=P\left(X=0,Y=2\right)+P\left(X=1,Y=2\right)+P\left(X=2,Y=2\right)$ $=0.1+0.1+0.1=0.3$ ，则 $E\left(Y\right)=0\times P\left(Y=0\right)+1\times P\left(Y=1\right)+2\times P\left(Y=2\right)$ $=0\times0.3+1\times0.4+2\times0.3=1$ ，则 $E\left(X-Y\right)=E\left(X\right)-E\left(Y\right)=1-1=0$ ， $P\left(X^2=0\right)=P\left(X=0\right)=0.4$ ， $P\left(X^2=1\right)=P\left(X=1\right)=0.2$ ， $P\left(X^2=4\right)=P\left(X=2\right)=0.4$ ，则 $E\left(X^2\right)=0\times P\left(X^2=0\right)+1\times P\left(X^2=1\right)+4\times P\left(X^2=4\right)$ $=0\times0.4+1\times0.2+4\times0.4=1.8$ ，则 $D\left(X\right)=E\left(X^2\right)-E^2\left(X\right)=1.8-1=0.8$ ， $P\left(Y^2=0\right)=P\left(Y=0\right)=0.3$ ， $P\left(Y^2=1\right)=P\left(Y=1\right)=0.4$ ， $E\left(Y^2\right)=0\times P\left(Y^2=0\right)+1\times P\left(Y^2=1\right)+4\times P\left(Y^2=4\right)$ $=0\times0.3+1\times0.4+4\times0.3=1.6$ ，则 $D\left(Y\right)=E\left(Y^2\right)-E^2\left(Y\right)=1.6-1=0.6$ ， $P\left(XY=0\right)=P\left(X=0\right)+P\left(Y=0\right)-P\left(X=0,Y=0\right)$ $=0.4+0.3-0.2=0.5$ ， $P\left(XY=1\right)=P\left(X=1,Y=1\right)=0$ ， $P\left(XY=2\right)=P\left(X=1,Y=2\right)+P\left(X=2,Y=1\right)$ $=0.1+0.3=0.4$ ， $P\left(XY=4\right)=P\left(X=2,Y=2\right)=0.1$ ，则 $E\left(XY\right)=0\times P\left(XY=0\right)+1\times P\left(XY=1\right)$ $+2\times P\left(XY=2\right)+4\times P\left(XY=4\right)$ $=0\times0.5+1\times0+2\times0.4+4\times0.1=1.2$ ，则 $Cov\left(X,Y\right)=E\left(XY\right)-E\left(X\right)E\left(Y\right)$ $=1.2-1\times1=0.2$ .