For
n
≥
0
n \geq 0
, let
λ
(
n
)
\lambda (n)
denote the median of the
Γ
(
n
+
1
,
1
)
\Gamma (n + 1,1)
distribution. We prove that
n
+
2
3
>
λ
(
n
)
≤
min
(
n
+
log
2
,
n
+
2
3
+
(
2
n
+
2
)
−
1
)
n + \tfrac {2}{3} > \lambda (n) \leq \min (n + \log 2, n + \tfrac {2}{3} + {(2n + 2)^{ - 1}})
. These bounds are sharp. There is an intimate relationship between
λ
(
n
)
\lambda (n)
and an equation of Ramanujan. Based on this relationship, we derive the asymptotic expansion of
λ
(
n
)
\lambda (n)
as follows:
\[
λ
(
n
)
=
n
+
2
3
+
8
405
n
−
64
5103
n
2
+
2
7
⋅
23
3
9
⋅
5
2
n
3
+
⋯
.
\lambda (n) = n + \frac {2}{3} + \frac {8}{{405n}} - \frac {{64}}{{5103{n^2}}} + \frac {{{2^7} \cdot 23}}{{{3^9} \cdot {5^2}{n^3}}} + \cdots .
\]
Let median
(
Z
μ
)
({Z_\mu })
denote the median of a Poisson random variable with mean
μ
\mu
, where the median is defined to be the least integer m such that
P
(
Z
μ
≤
m
)
≥
1
2
P({Z_\mu } \leq m) \geq \tfrac {1}{2}
. We show that the bounds on
λ
(
n
)
\lambda (n)
imply
\[
μ
−
log
2
≤
median
(
Z
μ
)
>
μ
+
1
3
.
\mu - \log 2 \leq {\text {median}}({Z_\mu }) > \mu + \frac {1}{3}.
\]
This proves a conjecture of Chen and Rubin. These inequalities are sharp.