Suppose we have a program that takes as input a number x on...

Suppose we have a program that takes as input a number $x$ on the real interval $[0,1]$ and outputs $f(x)$ for some bounded function $f$ taking on values in the range $[1, b]$. We want to estimate $$ \int_{x=0}^{1} f(x) d x $$ Assume that we have a random number generator that can generate independent uniform random variables $X_{1}, X_{2}, \ldots$. Show that $$ \sum_{i=1}^{m} \frac{f\left(X_{i}\right)}{m} $$ gives an ( $\varepsilon, \delta$ )-approximation for the integral for a suitable value of $m$.

Suppose we have a program that takes as input a number $x$ on the real interval $[0,1]$ and outputs $f(x)$ for some bounded function $f$ taking on values in the range $[1, b]$. We want to estimate
$$
\int_{x=0}^{1} f(x) d x
$$
Assume that we have a random number generator that can generate independent uniform random variables $X_{1}, X_{2}, \ldots$. Show that $$
\sum_{i=1}^{m} \frac{f\left(X_{i}\right)}{m}
$$
gives an ( $\varepsilon, \delta$ )-approximation for the integral for a suitable value of $m$.

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Key Concepts

Monte Carlo Integration

Monte Carlo Integration is a numerical method for approximating integrals by taking random samples of the integrand and averaging the results. This method is particularly useful when dealing with high-dimensional integrals or integrals with complicated integrands, as it converts the problem of numerical integration into one of statistical estimation.

Law of Large Numbers

The Law of Large Numbers is a foundational principle in probability theory that states that as the sample size increases, the sample average converges to the expected value. This concept justifies using the average of randomly sampled function values as an estimator for the integral, ensuring that with a sufficiently large number of samples, the approximation becomes accurate.

Concentration Inequalities

Concentration inequalities, such as Hoeffding's inequality, provide probabilistic bounds on how much the sample average deviates from its expected value. They are used to determine the sample size needed to achieve a desired level of accuracy (an (?, ?)-approximation), guaranteeing that the probability of the estimation error exceeding ? is at most ?.

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