What is the distribution of Y, where $$Y = \sum_{n=1}^{\infty}X_n\frac{1}{2^n},$$ with $$X_n \sim Bernoulli(1/2).$$ and independent. I think viewing things in binary numbers will help. Also it seems to be connected to Rademachers Function. My idea was a uniform distribution on the interval [0,1], but this is only my intuition and I am not able to show this rigorously.
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On $(0,10$ with Lebesgue measure define $Y_k(\omega)$ to be the $k-$th coefficient in the expansion of $\omega$ to base $2$. Somewhat tedious verification shows that $(Y_k)$ is i.i.d. with same success probability as $X_1$. Hence $\sum {x_k} /2^{k}$ has the same distribution as $\sum {Y_k} /2^{k}$ but the latter is the identity function on $(0,1)$. Hence the distribution is uniform. – Kavi Rama Murthy May 04 '20 at 23:49
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See also this question – saz May 05 '20 at 05:25