Question

若随机变量X与Y满足Y=1-X/2,且D(X)=2,则Cov(X,Y)= 答案是-1 请问为什么

Answer 1

CoV(X,Y)=E(XY)-E(X)E(Y)

将Y=1-X/2代入上式,有

CoV(X,Y)=E(X-X²/2)-E(X)E(1-X/2)=[E(X)-E(X²/2)]-E(X)(1-½E(X))=E(X)-½E(X²)-E(X)+½E(X)²

=½[E(X)²-E(X²)]=-½D(X)=-1

主要是公式E(X+Y)=E(X)+E(Y)以及D(X)=E(X²)-E(X)²的运用...