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I have an expression $f(x)$, outputting strictly real numbered values $\geq 0$ corresponding to the probability of some event, where $\sum_{i=0}^{N} f(i) = 1$. When is it true that $\int_{i=0}^{N} f(i) d(i) = 1$? If this isn't true, how do I find the average value of $f$, or points where $\sum_{i=0}^{r}f(i)=y$ for $0 \leq y \leq 1$?

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    @Argha My example fails regardless of $N$, you can set it to be infinite. I'd like to understand when this happens...?2012-12-22

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Mathematically $\sum$ and $\int$ are notation. We use $\sum$ for discrete(discontinuous) case and $\int$ for continuous case. Conceptually sometimes we use $\int$ instead of $\sum$ .

Let we find the sum $\sum_{k=1}^{n}k^{3/2}$ $\sum_{k=1}^{n}k^{3/2}=\int_{0}^{n}x^{3/2}~dx=\frac{2}{5}n^{5/2}$ Note that $\lim_{n \to \infty}\frac{1}{n}\sum_{r=1}^{n}f(\frac{r}{n})=\int_{0}^{1}f(x)~dx$ For large n, $\lim_{n \to \infty}\frac{1}{n}\sum_{r=1}^{n}(\frac{k}{n})^{3/2}=\int_{0}^{1}x^{3/2}~dx=\frac{2}{5}$ Then $\lim_{n \to \infty}\sum_{r=1}^{n}k^{3/2}=\frac{2}{5}n^{5/2}$

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    $\sum$ is not equivalent with $\int$. It only can be conceptually approximated by $\int$. For large $N$ the result given by $\sum$ is very near to the result given by $\int$ though they are not same. $\int$ value is always greater than $\sum$.2012-12-23