Prove that random variables are independent?

Question

If I have two random variables, say A and B, which are both discrete, can I prove that they are independent by proving that $E[A*B] = E[A]*E[B]$ ?

Answer 1

They are uncorrelated

$$ \begin{aligned} Cov(A,B) &= \mathbb{E}(AB) - \mathbb{E}(A)\mathbb{E}(B) \\ &= 0 \ \ (\because \ \ \mathbb{E}(AB) = \mathbb{E}(A)\mathbb{E}(B) \ \ ) \end{aligned} $$

since covariance is $0$, their Pearson correlation is $0$.

---

They can be dependent

Say

$$ P(A = -1) = P(A = 0) = P(A=1) = 1/3 \\ B= \begin{cases} 1 & \text{(if } A=0) \\ 0 & \text{otherwise}\end{cases} $$

then $\mathbb{E}(A) = 0, \mathbb{E} (AB) = 0$, but $B$ obviously depends on $A$.