The memory centre

Abstract

Let $x\in \mathbb{R}$ be given. As we know the, amount of bits needed to binary code $x$ with given accuracy ($h\in \mathbb{R}$) is approximately $ \m_{h}(x) \approx \log_{2}(\max {1, |\frac{x}{h}|}). $ We consider the problem where we should translate the origin $a$ so that the mean amount of bits needed to code randomly chosen element from a realization of a random variable $X$ is minimal. In other words, we want to find $a\in \mathbb{R}$ such that $$ \R \ni a \to \mathrm{E} (\m_{h} (X-a)) $$ attains minimum.