Standard Deviation of Nested Halving Intervals
Start with the interval $I_0 = [0, 1]$. At each step $i$, you pick a new interval $I_i$ uniformly at random inside $I_{i-1}$, where $I_i$ has exactly half the length of $I_{i-1}$ (so $I_i$ has length
^{-i}$).
The nested sequence $I_0 \supset I_1 \supset I_2 \supset \cdots$ converges to a single point $X \in [0,1]$.
What is the standard deviation of $X$?