Let $X_1,X_2,..., Xn$ be an i.i.d. sequence of binary random variables, each equally likely to be $0$ or $1$.
We define $Zn = 0$ for $n = 1$, and for $n>1$ as:
$Zn = |Xn −Xn_-1|$, where |.| denotes absolute value.
How can we find the probability distribution of $Z$ in order to calculate its entropy?
Can we argue that since $Xi$'s are i.i.d., therefore for any $Xn_-1$ the probability of $Xn$ not having the same value, is $0.5$ and thus $H(Z)=1$?