How to prove that if $x,y,z$ are random variables in $\Bbb{Z}_{257}$, then (x+y+z) is random.

Question

I would like to prove or disprove that if $x,y,z$ are three random variables in $\Bbb{Z}_{257}$, then (x+y+z) is random in $\Bbb{Z}_{257}$.
Added: Suppose $z,y,z$ are uniformly distributed in $\Bbb{Z}_{257}$, then $Pr(x)=\dfrac{1}{257}=Pr(y)=Pr(z)$. We have to calculate $Pr(x+y+z)=?$