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Let $X\sim\text{Binomial}(n,p)$ where $n$ is the number of trials and $p$ the probability of success of each trial. I am trying to evaluate the expected value of the following functions of $X$:

$$f(X)=\log[1+c(X-np)]$$ $$f^2(X)=(\log[1+c(X-np)])^2$$

Here $c$ is a very small positive constant (we can assume that $c<1/np$). We also have that $p\ll1$.

Are there closed form expressions for $E[f(X)]$ and $E[f^2(X)]$? Tight upper and lower bounds?

By Jensen's inequality and concavity of logarithm, $E[f(X)]\leq 0$.

I tried Taylor-expanding the logarithm and obtained expression involving the central moments of $X$, however, there is no closed form expression that I am aware for the central moments of binomial random variable.

Anyone have any other ideas?

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  • $\begingroup$ OP wrote: however, there is no closed form expression that I am aware for the central moments .... There should be no problem finding the central moments of a Binomial random variable. If you know how to find the raw moments, you can always use moment converter functions to express central moments in terms of raw moments. $\endgroup$ – wolfies Jul 29 '13 at 20:26

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