The memory centre

Let x 2 R be given. As we know the, amount of bits needed to binary code x with given accuracy (h2 R) is approximately mh(x) log 2 (maxf1;j x jg): We deal with the problem where we should put the origin a of the coordinate system 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 a2 R such that R3 a! E(mh(X a)) attains minimum. We show that under reasonable assumptions, asymptotically, the choice of a does not depend on h. Consequently, we reduce the problem to finding minimum of the function R3 a! Z R ln(jx aj)f(x)dx; where f is density distribution of the random variable X. Moreover, we provide constructive approach for determining a.