How to calculate the $\mathrm{Var}[X + Y]$ using covariance when $Y = X^2$. where $X$ can take the following values $\{-2,-1,0,1,2\}$. Each value can occur with equal probability i.e., $1/5$.
I know that following formula will be used $\mathrm{Var}[X+Y] = \mathrm{Var}[X] + \mathrm{Var}[Y] +2\text{cov}(X,Y)$. But i can't find a way to calculate $\mathrm{Var}[X]$ and $\mathrm{Var}[Y]$.
Calculation of $var (X)$ and $var(Y)$: $EX=0$ and $EX^{2}=\frac {4+1+1+4} 5=2$ so $var (X)=2$. $Y$ takes values $0,1,4$ with probabilities $\frac 1 5,\frac 2 5,\frac 2 5$ respectively. So $EY=2$ and $EY^{2}=\frac 2 5 +\frac 2 5 (16)=\frac {34} 5$. So $var (Y)=\frac {34} 5-4=\frac {14} 5$.