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## Summary:

- Understanding different notations of Monte Carlo approximations.

Currently working on a project involving Monte Carlo integrals. I haven't had any prior studies of this method, so hence the following question.

Consider the following expectation:

$$E[f(X)]=\int_A f(x)g(x)dx.$$

Let ##X## be a random variable taking values in ##A\subseteq\mathbb{R}^n##. Let ##g:A\to\mathbb{R}_+## be the probability density (pdf) of ##X##, and ##f:A\to\mathbb{R}## a function such that the expectation above is finite.

If ##X_1,X_2,...X_N## be independent random variables with pdf ##g##, then by the law of large numbers,

$$ E[f(X)]\to \frac{1}{N} \sum_{i=1}^N f(X_i) \quad \text{as N} \to \infty.$$

As far I understand, the sum above is a general Monte Carlo approximation of the integral.

Does the above approximation make any assumption on the pdf, i.e. uniformity and normalisation? If it is a general approximation, it should hold for any pdf, but I have seen different approximations such as ##V\frac{1}{N}\sum_{i=1}^N f(X_i)## and ##\frac{1}{N}\sum_{i=1}^N \frac{f(X_i)}{g(X_i)}##, where in the former ##V## denotes the definite integral over the pdf. How are these related and derived?

Consider the following expectation:

$$E[f(X)]=\int_A f(x)g(x)dx.$$

Let ##X## be a random variable taking values in ##A\subseteq\mathbb{R}^n##. Let ##g:A\to\mathbb{R}_+## be the probability density (pdf) of ##X##, and ##f:A\to\mathbb{R}## a function such that the expectation above is finite.

If ##X_1,X_2,...X_N## be independent random variables with pdf ##g##, then by the law of large numbers,

$$ E[f(X)]\to \frac{1}{N} \sum_{i=1}^N f(X_i) \quad \text{as N} \to \infty.$$

As far I understand, the sum above is a general Monte Carlo approximation of the integral.

Does the above approximation make any assumption on the pdf, i.e. uniformity and normalisation? If it is a general approximation, it should hold for any pdf, but I have seen different approximations such as ##V\frac{1}{N}\sum_{i=1}^N f(X_i)## and ##\frac{1}{N}\sum_{i=1}^N \frac{f(X_i)}{g(X_i)}##, where in the former ##V## denotes the definite integral over the pdf. How are these related and derived?

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