Entropy of a binomial distribution

How do we get the functional form for the entropy of a binomial distribution? Do we use Stirling's approximation?

According to Wikipedia, the entropy is:

$$\frac1 2 \log_2 \big( 2\pi e\, np(1-p) \big) + O \left( \frac{1}{n} \right)$$

As of now, my every attempt has been futile so I would be extremely appreciative if someone could guide me or provide some hints for the computation.

• A comment: the entropy of the normal distribution with variance $\sigma^2$ is ${1 \over 2} \log (2\pi e \sigma^2)$, which can be computed by a fairly straightforward integration. Perhaps using Stirling's approximation you can reduce the computation of the entropy of the binomial to this same integral plus some error terms. (I haven't actually tried to do this.) Nov 25, 2012 at 19:51

We are interested in the sum $$H = -\sum_{k=0}^n {n\choose k}p^k(1-p)^{n-k} \log_2\left[{n\choose k}p^k(1-p)^{n-k} \right].$$ For $n$ large we can use the de-Moivre-Laplace theorem, $$H \simeq -\int_{-\infty}^\infty dx \, \frac{1}{\sqrt{2\pi}\sigma} \exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right] \log_2\left\{\frac{1}{\sqrt{2\pi}\sigma} \exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right] \right\},$$ where $\mu = n p$ and $\sigma^2 = n p(1-p)$. Thus, $$\begin{eqnarray*} H &\simeq& \int_{-\infty}^\infty dx \, \frac{1}{\sqrt{2\pi}\sigma} \exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right] \left[\log_2(\sqrt{2\pi}\sigma) + \frac{(x-\mu)^2}{2\sigma^2} \log_2 e \right] \\ &=& \log_2(\sqrt{2\pi}\sigma) + \frac{\sigma^2}{2\sigma^2} \log_2 e \\ &=& \frac{1}{2} \log_2 (2\pi e\sigma^2) \end{eqnarray*}$$ and so $$H \simeq \frac{1}{2} \log_2 \left[2\pi e n p(1-p)\right].$$ Higher order terms can be found, essentially by deriving a more careful (and less simple) version of de-Moivre-Laplace.
• Why is the "dx" before the term being integrated. I know that it is a multiplication, but doesn't convention put the increment of integration at the far right side of the integral? $\int{sin \left( x \right) \cdot dx}$ vs. $\int{dx \cdot sin \left( x \right) }$. Is there a reason for the difference? Mar 9, 2016 at 15:54
• @EngrStudent: I tend to think about integration as an operation that acts to the right like differentiation, rather than from outside to inside as it is often written: $\int_{y_1}^{y_2} dy\int_{x_1}^{x_2} dx(\ldots)$ rather than $\int_{y_1}^{y_2}\int_{x_1}^{x_2}(\ldots)dx\,dy$. This is probably due to my background in physics, where this notation is common. Mar 11, 2016 at 0:11