Using the Median to Estimate a Parameter I am trying to use the Median to estimate the value for the parameter $a$ in the following PMF.
\begin{equation}
\label{eq1:givenpdf}
    \mathbb{P}\left[X=\frac{a}{n}\right] = \frac{36}{5}\frac{n^2}{\left(n+1\right)\left(n+2\right)\left(n+3\right)\left(n+4\right)}\ for\ n=1,2,\dots
\end{equation}
When equating the CDF to 0.5 to obtain a formula for the median, the following was obtained:
\begin{equation}
\label{eq:Median}
    \implies \frac{a(5a^2+9an+4n^2)}{5(a+2n)(a+3n)(a+4n)} = 0.5 
\end{equation}
Solving for $n$ yielded:
$$n = -0.3389a - (0.1687 \pm 0.2923i)a + (0.0590 \mp 0.1023i)a \approx (-0.5667 \pm 0.1900i)a$$
$$ \text{and}$$
$$ n= 0.0056 \left[-61a + 60.7433a + 21.2534a \right] \approx 0.1167a$$
For which I chose the real root, for obvious reasons.
How can I use this information to obtain an Estimator for $a$ using the median and apply it on the data set attached below (with is summary statistics)?
I tried researching online, however I am still getting confused because the median for the data set provided is $0.5$



The median of this distribution is at $a/9$, since
$$P[X<\frac{a}9]=48.6\%$$
$$P[X=\frac{a}9]=\ \ 3.4\%$$
$$P[X>\frac{a}9]=48.0\%$$
So you can estimate $a$ as $9$ times the median.
