# Variance and Convergence

## Definition of Variance

$$\text{Var}(X) = E\{(x - \mu)^2\}$$

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## Variance of the Monte Carlo Estimator

$$\text{Var}(\hat{I}) = \frac{(b-a)^2}{n} \text{Var}(f(x))$$

### Convergence Rate

Standard deviation scales as:
$$\sigma \sim \frac{1}{\sqrt{n}}$$

This is good for a small number of dimensions.

**However:** With higher dimensionality, variance gets exponentially worse (curse of dimensionality).

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## Multi-Dimensional Case

Combine with Monte Carlo Integration techniques (importance sampling, stratified sampling, etc.) to manage variance in high dimensions.