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 $\sqrt{π/\,2}$. 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.