急用!由E(xy)=E(x)E(y)能否推出D(xy)=D(x)D(y)?如果不能,那又能推出什么有关D的结论?

Question

急用!由E(xy)=E(x)E(y)能否推出D(xy)=D(x)D(y)?
如果不能,那又能推出什么有关D的结论?

Answer 1

只能推出D(X±Y)=D(X)+D(Y)
D(X)=E(X²)-E²(X)
D(X±Y)=E[(X±Y)²]-E²(X±Y)=E(X²)±2E(XY)+E(Y²)-[E²(X)±2E(X)E(Y)+E²(Y)]
=[E(X²)-E²(X)]+[E(Y²)-E²(Y)]±2E(XY)-[±2E(X)E(Y)]
=D(X)+D(Y)±2E(XY)-[±2E(X)E(Y)]
代入E(XY)=E(X)E(Y)得到：
D(X±Y)=D(X)+D(Y)