Computing Gamma Median by Hand?

Question

I am stuck computing median of Gamma distribution by hand. Let $X$ have gamma distribution with $\alpha = 6$ and $\beta = 2$. I know how to compute the median of this distribution using numerical methods in R, but I am unsure how to compute this quantity by hand. All I know is that the gamma poisson relationship must be invoked. Thus, we could compute the median by solving $1 - P(Y < \alpha) = .5$, where $Y$ ~ $Poisson(x^{*} / \beta)$, with $x^{*}$ representing the median of the gamma distribution. Expanding the Poisson CDF gets really messy, and solving for $x^{*}$ seems really difficult, so any help computing the median by hand would be appreicated. Thanks Stack.

Answer 1

In [1], a sharp bound on the median of gamma distribution is introduced as $$ n+\frac{2}{3} < median(n) < min(n+log2 , n+\frac{2}{3}+(2n+2)^{-1}) $$ in which the bound decrease with n, the parameter of Gamma distribution $\Gamma(n+1,1) $. It is also mentioned that the asymptotic expansion of median could be derived as $$ median(n) = n + \frac{2}{3} + \frac{8}{405n} - \frac{64}{5103n^2} + ... $$ For more information, it's best to review the reference.

[1]: Choi, K. P., On the medians of gamma distributions and an equation of Ramanujan, Proc. Am. Math. Soc. 121, No. 1, 245-251 (1994). ZBL0803.62007.