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 $0 \leq y \leq 1$?
Integral versus a sum for an expression that sums to one
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calculus
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0you wanna talk about $\int_{i=0}^{N} f(i)~di = 1$ ? – 2012-12-22
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0@Argha Yes, that's right. – 2012-12-22
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0Since your sum is **1** you can use $\int$ instead of $\sum$ for moderate large **n**. – 2012-12-22
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0@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