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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

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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}

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