Proving consequence of $\operatorname{var}(X)+\operatorname{var}(Y)=\operatorname{var}(X+Y)$

Question

How to prove that if $\operatorname{var}(X)+\operatorname{var}(Y)=\operatorname{var}(X+Y)$, then $X$ and $Y$ are independent?

Answer 1

That is false. Let $X_0=X-E(X)$ and $Y_0=Y-E(Y)$. Then

$$V(X+Y)=V(X_0+Y_0)=E((X_0+Y_0)^2) = V(X_0)+V(Y_0)+2 E(X_0 Y_0).$$

The last term is the covariance. It is zero for independent variables but it can be zero for dependent variables as well.

For a wealth of examples, suppose $X$ is any random variable which is symmetric about $0$ (for example a $N(0,1)$ variable). Let $Y=X^2$. Then $X$ and $Y$ are dependent but their covariance is $E(X^3)=0$ by the symmetry.

Answer 2

This is not true.

Consider $$X \sim U[-1,1]$$ and $Y = -X$ if $X\leq 0$ and $Y = X$ if $X\geq 0$.

Answer 3

As the other answers indicate, this is not true. The correct conclusion is that the variables are uncorrelated, not that they are independent. These are not the same thing.

See https://web.archive.org/web/20171110154645/http://mathforum.org/library/drmath/view/69928.html

Answer 4

Well if $X$ and $Y$ are jointly normal, then they are also independent.