$W$ is a random variable taking values $q_i=e^{-\frac{1}{i+\alpha}}$ with probability $P(W=q_i)=\left(\begin{array}{l} n \\ i \end{array}\right) p^{i}(1-p)^{n-i}=p_{i}$.
What is the value of $p$ in terms of $\alpha$ which maximizes variance of $W$ denoted by $\sigma_W^2$.? (Here $\alpha >0$.)
I am able to numerically solve this for a given value of $\alpha$. But, I am more interested in an analytical solution even if it is an approximate one. Can someone help me solve this or suggest possible ways to consider in solving this.