A good random number generator, $G$ will produce a sequence of $[0, 1]$ values which are near uniformly distributed as $n$ draws goes to infinity.
If I start drawing samples from the random number generator $G$ and I add a small $\epsilon$ to each value (noise), where $\epsilon$ may be biased in an unknown way, I want to compute a correction $\delta$ which adjusts for any drift produced by the $\epsilon$ noise such that my sequence still approaches the uniform distribution as $n \to \infty$.
Suppose the biased samples are given by $(G_i)_{i\in\mathbb N}$. Then recursively define a new sequence $(H_i)_{i\in\mathbb N}$ by setting $H_1=G_1$ and $$ H_n=G_n-\frac{H_1+\cdots+H_{n-1}}{n-1}+\frac{1}{2}. $$ (Note that for small $n$, the $H_n$ can fall outside the interval $[0,1]$ with low probability. You may want to clamp it to $[0,1]$, which will have negligible effect in the large $n$ limit.)