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I am looking for a concise description to help me understand the error for Monte Carlo Integration using Uniform Sampling and Importance Sampling

For Importance Sampling I have that the error is just the square root of the variance...

$$\sqrt{\frac{1}{N}<f(Xi)^2>-<\frac{1}{N}f(Xi)>^2)}$$

for Uniform Sampling what would the error be similar such as

$$\frac{b-a}{\sqrt{N}}\sqrt{\frac{1}{N}<f(Xi)^2>-<\frac{1}{N}f(Xi)>^2)}$$

where $a$ and $b$ are the limits of the integration??

This to me isn't right as my understanding was that an importance sampling would decrease the variance of the MC integration.

I can't find any clear explanation of the difference online so any help is appreciated.

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    $\begingroup$ I can't really answer your question, but it seems odd to me that there are no integration bounds in your first formula. Don't you need them in any case, no matter the sampling strategy? $\endgroup$
    – MPIchael
    Oct 23 '20 at 7:54
  • $\begingroup$ In both of the formulas you show, there is a factor $1/N$ that appears once within the squares and once outside the squares. This seems wrong to me. $\endgroup$
    – Wolfgang Bangerth
    Oct 23 '20 at 14:15
  • $\begingroup$ Your question would be clearer if you cited the sources of these formulas. $\endgroup$
    – Brian Borchers
    Oct 27 '20 at 4:28

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