I'm currently looking through Oliver Knill's book Probability Theory and Stochastic Processes and am trying to understand Knill's proof that the variance of a random variable $X$ with the Cantor distribution is 1/8 (Corollary 3.1.6., pp. 129-130).
Taking $X$ to be a r.v. with the Cantor distribution and $Y = 1_{[0,1/3]}$, we have by the law of total variance that
$V[X]=E[V[X|Y]]+V[E[X|Y]]$.
Knill provides an argument that $E[V[X|Y]]=\frac{1}{9}V[X]$, which is the part I don't understand. The specific part that throws me is his assertion that (with $Z=E[X|Y]$),
$V[X|Y]=E[X^2|Y]-E[X|Y]^2=E[X^2|Y]-Z$.
If anyone could provide some insight into how the last equality above works (i.e., how $E[X|Y]^2=Z$) or more generally an explanation of how $E[V[X|Y]]=\frac{1}{9}V[X]$, I'd really appreciate it.