Lower bound functional binomial r.v.

Question

I am trying to find a bound of the type

$\mathbb{E}(|B-\frac{N}{2}|) \geq C \sqrt{N}$

Where $B$ is a binomial variable with parameters $(N,\frac{1}{2})$.

The bound doesn't need to be very tight in the constant, however it is important for me that $C$ is explicit. It cannot be just an asymptotic bound. It should be valid for every $N$.

Thanks.

Answer 1

Better than an approximate lower bound, ... one can calculate the desired expectation exactly (and neatly too). In this instance, $X \sim Binomial(n,\frac12)$ with pmf $f(x)$:

Then, the expectation you seek is:

Here is a plot of the solution, as parameter $n$ varies:

Notes

1. The Expect function used above is from the mathStatica package for Mathematica. As disclosure, I should add that I am one of the authors.
2. Ceiling[x] rounds up to the nearest integer; e.g. Ceiling[2.2] returns 3.