Are discrete random variables, with same domain and uniform probability, independent?

Question

If X and Y are discrete random variables, both with domain {-1,0,1}, with uniform probability, does this imply they are independent? What would be the expected value E(XY) and probability like P(XY=1) or P(XY=-1) in this case?

Answer 1

does this imply they are independent?

No! To get independence you have to use the corresponding definition:

$$f_{XY}(x,y)=f_X(x)\cdot f_Y(y)$$

$\forall (x,y)$

In your case you do not have enough information to conclude for independence. You need, for example, their joint distribution.

Here are two examples with the same uniform marginals which are independent (example 1) and not independent (example 2)

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To calculate correlation coefficient, first calculate covariance

$$\text{Cov}[X,Y]=\mathbb{E}][XY]-\mathbb{E}[X]\mathbb{E}[Y]=\Sigma_i\Sigma_j XiY_i p(x_i,y_j)-0=$$

$$=-1\cdot(-1)\cdot\frac{1}{6}+(-1)\cdot0\cdot\frac{1}{6}+\dots=\frac{1}{6}$$

thus

$$\rho_{XY}=\frac{\frac{1}{6}}{\frac{1}{3}}=\frac{1}{2}$$