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

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