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I have a discrete random variable $X$ with $P(X \geq x) = c^x$ and I would like to bound $E(\log{X})$. I can write this as follows I think $E(\log{X}) = \sum_{x=1}^{\infty} c^x \log{x}.$

We know that $0\leq c \leq 1$. I would like to bound $E(\log{X})$ above and below.

One would approach would be to replace the sum by an integral but I didn't get anywhere. Can anyone see how to get good bounds?

Question has been edited to make it clearer.

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    If $P(X\ge x)=c^x$, then $X=0$ does happen since $P(X=0)=1-c$. This is why I modified this hypothesis in my answer.2012-11-13

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Let us assume that $X\geqslant1$ is geometric with parameter $a$ in $(0,1)$, that is, that, for every $n\geqslant1$, $\mathbb P(X\geqslant n)=(1-a)^{n-1}$. Then $\mathbb E(X)=1/a$, hence Jensen inequality yields $ \mathbb E(\log X)\leqslant\log \mathbb E(X)=-\log a. $ On the other hand, the function logarithm is nondecreasing hence, for every $n\geqslant1$, $ \mathbb E(\log X)\geqslant\log(n)\cdot \mathbb P(X\geqslant n)=\log(n)\cdot (1-a)^{n-1}. $ In particular, for $n$ the integer part of $1/a$, one gets approximately $ \mathbb E(\log X)\stackrel{(\mathrm{approx.})}{\geqslant}-\log(a)\cdot(1-a)^{(1-a)/a}. $ Note that when $a\to0$, $(1-a)^{(1-a)/a}\to\mathrm e^{-1}$ hence the lower bound is asymptotically of the order of the upper bound.

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    I would vote it up if I could. Thanks!2012-11-13