I have joint probability of two variables X and Y,
\begin{array}[] \hline & y=-1 & y=0 & y=1 & y=2 &P(X=x_i) \\ x=0 & 0.06 & 0.25 & 0.02 & 0.02 & 0.35\\ x=1 & 0.31 & 0.02 & 0.27 & 0.02 & 0.62\\ x=2 & 0.018 & 0 & 0.006 & 0.006 & 0.03\\ P(Y=y_i) & 0.388 & 0.27 & 0.296 & 0.046 & 1 \end{array}
I calculated their marginal dependencies and added it to the above last row and column respectively.
I calculated the expectation value using the formula,
$E(X)=\sum_{i=0}^{2}(X_{i}*P(X_{i}))$
and calculated $E(X)=0.68$ and $E(Y)=0$, while calculating $E(XY)$, I need to chose the formula based upon the dependency of X and Y.
How can i deduce the dependency?.
The random variables are not independent. Probably the simplest way to do this is to note that $\Pr(X=2\cap Y=0)=0$, while neither $\Pr(X=2)$ nor $\Pr(Y=0)$ is $0$. The random variables would be independent if and only if for every $x$ and $y$, $\Pr(X=y\cap Y=y)=\Pr(X=x)\Pr(Y=y)$.
But you may want to find out whether the random variables are correlated or not. (If they are independent, they are uncorrelated, but it is possible for random variables to be uncorrelated but not independent.)
The random variables are uncorrelated precisely if $E(XY)=E(X)E(Y)$. So you need to compute $E(XY)$. I will assume you know how to do that. Please leave a message if that poses difficulty.
If they are correlated, you may wish to compute the correlation coefficient. This is $\frac{E(XY)-E(X)E(Y)}{\sqrt{\text{Var}(X)\text{Var(Y)}}}$. Additional calculation will be needed to find the variances.