Suppose $X_n$ and $Y_n$ are iid variables, each distributed binomially as $B(n,1/2)$.
Let $M_n$ be the median of $2^{X_n}+2^{Y_n}$. Empirically
$$\sqrt{2}\, \le \left(\frac{M_n}{2}\right)^{1/n}\!\! \le \frac32$$
What can we say about $\lim_{n\to\infty}\left(\frac{M_n}{2}\right)^{1/n}$, whose graph looks like the below?
- Does the limit exist?
- Is the limit properly between $\sqrt{2}$ and $3/2$?
- Does the limit have a nice formula?
The sequence is in the OEIS as A288347, which has an interpretation, methods of calculation, and some calculated values.
The first inequality above says that $\text{median}(2^{X_n})+\text{median}(2^{Y_n})\le \text{median}(2^{X_n}+2^{Y_n})$, which looks easy enough to prove.
The second inequality above says that $\text{median}(2^{X_n}+2^{Y_n}) \le \text{mean}(2^{X_n}+2^{Y_n})$, which is a discretized special case of another question, and looks hard to prove.