# probability distribution of a sum of random variables [closed]

Suppose we have a random variable $$X$$ $$P[X=-1]=1/3$$, $$P[X=0]=1/3$$ and $$P[X=1]=1/3$$

now let $$Y=X^2$$

we have $$n$$ independent realizations of $$Y$$ $$(Y_1, Y_2,......, Y_n)$$ what is the probability distribution of these observations?

now let $$Z=Y_1+Y_2+.........+Y_n$$

what is the probability distribution of $$Z$$?

I know $$X$$ is a uniform distribution but I don't know what is the distribution of a uniform squared and how to find the density of multiple observations or the density of a sum

Hint $$X_i^2 \sim Ber(\frac{2}{3})$$ so $$\sum X_i^2 \sim Bin(n,\frac{2}{3})$$
$$\begin{array}{c|ccc} X & -1 & 0 & 1 \\ \hline P & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ X^2 & 1 & 0 & 1 \end{array}$$