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.
Categories: math.PR math.HO