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

The rules of a popular dice game 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 $n$. The expected value of the gain, $\mathbb{E}\,[\,G_n\,]$, exists for all $n$. A numerical study suggests that, in the limit of $n \to \infty$, the expectation of the scaled gain $\mathbb{E}\,[\,H_n\,]=\mathbb{E}\,[\,G_n/\sqrt{n}\,]$ converges to a constant which is empirically conjectured to be equal to $\sqrt{\pi/\,2}$. In order to prove this conjecture, we derive an analytic expression of the expected gain $\mathbb{E}\,[\,G_n\,]$ and show that its asymptotic behaviour for $n\to\infty$ implies indeed convergence of $\mathbb{E}\,[\,H_n\,]$ to $\sqrt{\pi/\,2}$. An analytic expression of the variance of the gain $G_n$ is derived by a similar technique. Its asymptotic behaviour for $n\to\infty$ implies that the variance of $H_n$ converges to $2\!-\!\pi/\,2$. Finally, it is proved that $H_n$ converges weakly to the Rayleigh distribution with scale parameter 1, mean $\sqrt{\pi/\,2}$ and variance $2\!-\!\pi/\,2$.