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

The rules of a game of dice are extended to a hyper-die'' with
n
∈
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
→
∞
the expectation of the scaled gain
E
[
H
n
]
=
E
[
G
n
/
√
n
]
converges to
√
π
/
2
.

The conjecture is proved by deriving an analytic expression of the expected gain
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.