已知随机变量X和Y的方差为D(X)=1,D(Y)=4,Cov(x,y)=1,记U=X-2Y,V=2X-Y已知随机变量X和Y的方差为D(X)=1,D(Y)=4,Cov(X,Y)=1,记U=X-2Y,V=2X-Y,试求Cov(U,V)和ρUV.求过程,谢谢!
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D(a)=2²*D(x)+D(y)+2*2*(-1)*cov(x,y)=4D(x)+D(y)-4cov(x,y)=4+4-4=4
D(b)=D(x)+(-2)²*D(y)+2*(-2)*1*cov(x,y)=D(x)+4D(y)-4cov(x,y)=1+16-4=13
cov(a,b)=E(ab)-E(a)*E(b)=E(2*x²-5xy+2y²)-E(2x-y)E(x-2y)
=2E(x²)-5E(xy)+2E(y²)-2(E(x))²+5E(x)E(y)-2(E(y))²
=2( E(x²)-(E(x))² ) + 2( E(y²)-(E(y))² ) - 5( E(xy) - E(x)E(y) )
=2D(x)+2D(y)+5cov(x,y)=2+8-5=5
∴ρ(a,b)=cov(a,b)/√(D(a)D(b))=5/(2*√13)