Hoi, i dont see why the following is true

Let $\displaystyle \mathbb{P}(X_k =1)=\mathbb{P}(X_k=0)=1/2$ ( i.i.d. Bernoulli trials)

and consider $\displaystyle Y = 3\sum_{k=1}^{\infty}4^{-k}X_k $ ( Then clearly $\displaystyle 0\leq Y \leq 1 $)

Apparantly the distribution function F of Y is constant on the interval $\displaystyle (1/4,3/4)$ and satisfies

$\displaystyle F(x) = 1-F(x)$ and

for $\displaystyle x< 1/4$ it satisfies $\displaystyle F(x)=2F(x/4)$

I'm a bit bewildered...:/