A new discrete distribution arising from a generalised random game and its asymptotic properties

TL;DR

This paper introduces a new discrete distribution from a generalized dice game, deriving its probability functions, analyzing its asymptotic behavior, and proving convergence to the Rayleigh distribution.

Contribution

It provides explicit formulas for the distribution of the game gain and proves its asymptotic convergence to a Rayleigh distribution, extending understanding of such probabilistic models.

Findings

Expected scaled gain converges to √(π/2).

Variance of the gain is explicitly derived.

Scaled gain converges weakly to Rayleigh distribution.

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.