Decomposition of variance

Question

Suppose X is a continuous random variable that can take any value between plus and minus infinity. Furthermore, suppose A is a random variable capturing those events where X is below 0, and B is a random variable capturing those events where X is above 0.

Is there a general relationship between variance(X), variance(A), and variance(B)?

UPDATE: Siong and Canardini provide the same answer. Unfortunately, my simulation efforts do not agree, so I wonder where the mistake is:

In the following I draw 10 numbers (MATLAB, randn) and try to implement the given answers. What am I doing wrong?

Answer 1

$$X=A+B$$

$$Var(X)=Var(A)+Var(B)+2Cov(A,B)$$

$$Cov(A,B)=\mathbb{E}[AB]-\mathbb{E}[A]\mathbb{E}[B]=-\mathbb{E}[A]\mathbb{E}[B]\geq 0$$

$$Var(X)=Var(A)+Var(B)-2\mathbb{E}[A]\mathbb{E}[B]$$

$$Var(A)+Var(B) \leq Var(X)$$

Answer 2

We have $$Var(X)=Var(A)+Var(B)+2Cov(A,B)$$

$$Cov(A,B)=E(AB)-E(A)E(B)$$

Given that $AB=X^21_{X \leq 0}1_{X >0}$, we have $AB=0$ a.s, $\{X \leq 0\}$ and $\{X >0\}$ being incompatible.

Thus, $$Var(X)=Var(A)+Var(B)-2E(A)E(B)$$