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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.

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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.) – Michael Lugo Nov 25 '12 at 19:51
up vote 10 down vote accepted

This answer follows roughly the suggestion of @MichaelLugo in the comments.

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.

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Thanks!You really helped me. – user844541 Nov 26 '12 at 7:16
@user844541: You are welcome! – user26872 Nov 26 '12 at 11:56
How is the integration done in the second to last step? – Thomas Ahle Jul 15 '14 at 18:55
@user26872 is there any analogue for multinomial distribution? Thanks in advance! – Egorova Lena Jan 14 at 13:28
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? – EngrStudent Mar 9 at 15:54

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