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