Let $X$ be an integer sampled uniformly from the (integer) range $\{ 0...2^k - 1 \}$ .

Now, consider the distribution of

$\#\text{bits}(X) = \begin{cases} 1 & X = 0 \\\\ 1 + \lfloor \log_2(X) \rfloor & \text{otherwise} \end{cases}$


$\#\text{bytes}(X) = \begin{cases} 1 & X = 0 \\\\ 1 + \lfloor \log_{256}(X) \rfloor & \text{otherwise} \end{cases}$

which are the numbers of bits and of bytes respectively necessary to hold $X$.

My question: Is there some commonly-used term for the one of the distributions of $\#\text{bits}(X)$ or $\#\text{bytes}(X)$ ? Or for a distribution which is very similar to them?

  • $\begingroup$ Geometric distribution. $\endgroup$ – Yuval Filmus Dec 23 '18 at 21:41
  • $\begingroup$ @YuvalFilmus: Not exactly, but that's super-close. Thanks and see my answer. $\endgroup$ – einpoklum Dec 23 '18 at 23:36

As @YuvalFilmus suggests, these distributions are very close to being geometric. They aren't geometric, since they're bounded, which is what threw me off, but think about it this way:

A geometric distribution corresponds to the number of Bernouli trials up to and including a first success (in some formulations: not including the first success). The distribution parameter p is the probability of succeeding in each individual experiment. Now,

  • Our first "experiment" involves sampling the most significant bit (or byte): Will it be 0, or something else? A 0 is a success, something else is a failure.
  • Our second "experiment" is the same, but for the second-most-significant bit (or byte).
  • and so on.

In a geometric distribution proper, experiments never cease until the first success. In our case, $k$ experiments are conducted, and if you've failed them all then you're simply at 0, with probability $2^{-k}$.


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