This question came up in a course on measure theoretic probability theory. We have had lots of information on the existence of distribution, but no examples of how to find/construct them. Here's the problem:
Let $X_1,X_2,\dots$ be an $iid$ sequence of Bernoulli random variables, defined on $(\Omega,\mathcal{F},\mathbb{P})$ with $\mathbb{P}(X_1=1)=\frac{1}{2}$. Let
$X = \sum_{k=1}^\infty2^{-k}X_k.$
Find the distribution of $X$.
Clearly $F(x) = \mathbb{P}(X \leq x) = \int_{-\infty}^xX\,d\mathbb{P}$, but how to go from hear I have no idea. I also tried the path $P(\sum_{k=1}^\infty2^{-k}X_k \leq x) = \mathbb{P}(X = 1)\mathbb{P}(\sum_{k=2}^\infty2^{-k}X_k\leq x - \frac{1}{2}) + \mathbb{P}(X=0)\mathbb{P}(\sum_{k=2}^\infty2^{-k}X_k \leq x)$ but this seems to be a dead end.
Any tips tips how to tackle this?