I am interested in the entropy of the binomial random variable $Binom(n,1-2^{-1/n})$.
Specifically, I would like to show that this is an increasing function in $n$.?
I was able to verify this through numerical simulation. Also, if the probability parameter was a constant (not a function of $n$) it is known to be an increasing function in $n$ as shown here.
Can someone help me prove this property.
My attempt:
From this paper, we know that entropy of $Binom(n,\lambda/n)$ is an increasing function in $n$.
Also, entropy of $Binom(n,p)$ is maximum when $p=1/2$.
For large values of $n$, $Binom(n,1-2^{-1/n})=Binom(n,\frac{\ln 2}{n})$. Thus, once $n$ is large enough, entropy of $Binom(n,1-2^{-1/n})$ is increasing function in $n$.
But I am still not sure how to establish the conjecture for low values of $n$. Any inputs are welcome.
