A New Discrete Distribution Arising from a Generalised Random Game and Its Asymptotic Properties

Abstract

The rules of a game of dice are extended to a ``hyper-die'' with \(n\in\mathbb{N}\) equally probable faces, numbered from 1 to \(n\). We derive recursive and explicit expressions for the probability mass function and the cumulative distribution function of the gain \(G_n\) for arbitrary values of \(n\). A numerical study suggests the conjecture that for \(n \to \infty\) the expectation of the scaled gain \(\mathbb{E}[{H_n}]=\mathbb{E} [{G_n/\sqrt{n}\,}]\) converges to \(\sqrt{\pi/\,2}\).
 The conjecture is proved by deriving an analytic expression of the expected gain \(\mathbb{E} [{G_n}]\).
 An analytic expression of the variance of the gain \(G_n\) is derived by a similar technique. Finally, it is proved that \(H_n\) converges weakly to the Rayleigh distribution with scale parameter~1.