Are discrete random variables, with same domain and uniform probability, always independent? 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 $\mathbb{E}(XY)$ and probability like $\mathbb{P}(XY=1)$ or $\mathbb{P}(XY=-1)$ in this case?
 A: You can say $X$ and $Y$ each have mean $0$ and variance $\frac23$ and in general

*

*Their covariance is equal to $\mathbb E[XY]$ here and can be any value between $-\frac23$ and $+\frac23$ so their correlation can be any value between $-1$ and $+1$

*$\mathbb P[XY=+1]$ and $\mathbb P[XY=-1]$ can each be between $0$ and $\frac23$

*$\mathbb P[XY=0]$ can be between $\frac13$ and $\frac23$
If they are independent then

*

*Their covariance is equal to $\mathbb E[XY]$ and is $0$

*$\mathbb P[XY=+1]=\mathbb P[XY=-1] =\frac29$

*$\mathbb P[XY=0] =\frac59$
It is possible for them to have covariance and $\mathbb E[XY]=0$ without them being independent, for example if $(0,0)$ has probability $\frac13$ while $(+1,+1)$, $(+1,-1)$, $(-1,+1)$, $(-1,-1)$ each have probability $\frac16$, in which case $\mathbb P[XY=+1]=\mathbb P[XY=-1] =\frac13$  and also $\mathbb P[XY=0] =\frac13$
A: No, suppose you have $X$ as you described and for $Y$ you flip a coin and in 0.5 probability it takes the value of $X$ otherwise you uniformly sample from $\{-1,0,1\}$.
The marginal probabilities will be uniform yet they are obviously dependent. Short simulation below.
x    <- sample(c(-1, 0, 1), 100000, TRUE)
coin <- rbinom(100000, 1, 0.5)
y    <- coin * sample(c(-1, 0, 1), 100000, TRUE) + (1 - coin) * x

     
> cor(x, y)
[1] 0.4986911


You can see that $X$ any $Y$ are dependent and have the same marginal uniform distribution.
A: It is simple to construct an example where both variables are marginally uniformly distributed, but they are not independent.  The simplest example is to take $X \sim \text{U} \{ -1,0,1 \}$ and let $Y=X$.  In this case both of the variables have a uniform distribution, but they are perfectly correlated.
