Understanding the meaning of this probability distribution

Solution 1:

The distribution you have is known as Zipf's distribution, formally speaking, for $s>1$ we have $$P(I\subseteq \mathbb{N})=c_s\sum_{k\in I}\frac{1}{k^{s+1}}$$ where $$c_s=\frac{1}{\displaystyle \sum_{k=1}^\infty\frac{1}{k^{s+1}}}$$ reason by wich it is also known as Zeta or Riemann's distribution, since $c_s$ is expressed as a fraction of the Riemann's function. So the necesity of taking $s>1$ is due to the convergence of the Riemann's function.

Applications of this distribution can be enumerated in ranking areas as linguistic(Zipf), family incomes in a given country(Pareto) and many others. You can read about it in Univariate Discrete Distributions of Samuel Kotz, from Wiley Series in Probability and Statistics.

Also i advice you to take a look at Zipf's Law, although it is a little harder.