The memory centre

Let x ∈ R be given. As we know the, amount of bits needed to binary code x with given accuracy (h ∈ R) is approximately mh(x) ≈ log2(max{1, | 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 ∈ R such that R 3 a→ E(mh(X − a)) attains minimum. We show that under reasonable assumptions, the choice of a does not depend on h asymptotically. Consequently, we reduce the problem to finding minimum of the function R 3 a→ ∫ R ln(|x− a|)f(x)dx, where f is the density distribution of the random variable X . Moreover, we provide constructive approach for determining a.