##  [Monte Carlo Integration](/monte-carlo-integration-0) 

 Definition

A numerical method that estimates definite integrals by averaging the values of the integrand evaluated at randomly sampled points from a chosen probability distribution over the domain.

 

 

 

 

 

 





## Principle

Principle

Use random sampling so that the sample mean of integrand evaluations converges to the integral by the law of large numbers; variance controls accuracy.

 

 

 

 

 





## Demonstration

Demonstration

Estimate the integral of f(x)=exp(-|x|) over a high-dimensional unit cube by drawing N independent uniform samples and computing the average f(x); increase N to reduce Monte Carlo error ≈ O(N^{-1/2}).

 

 

 

 

## Misapplication

Misapplication

Applying naive random sampling in very high dimension without variance reduction (importance sampling, stratification) or using too small N, yielding misleadingly noisy estimates.

 

 

 

 

 





## Consequence

Consequence

Provides dimensionally scalable approximation where deterministic quadrature is infeasible; error diminishes stochastically and can be quantified by sample variance.

 

 

 

 

## Reversal

Reversal

Deterministic quadrature or interpolation methods yield faster convergence for smooth low-dimensional integrands but typically fail to scale to very high dimensions.

 

 

 

 

 





## Boundary

Boundary

Requires the integrand to be integrable under the sampling law; not suitable when the integrand has infinite variance under the proposal or when exact symbolic integration is available and preferred.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Contrasts with deterministic cubature and with quasi-random (low-discrepancy) sequences; tension is between stochastic O(N^{-1/2}) convergence and faster rates for structured deterministic methods.

 

 

 

 

 





## Synthesis

Synthesis

Monte Carlo integration approximates integrals by random sampling and averaging; it trades statistical convergence speed for robustness to dimension and complicated domains, with accuracy governed by variance control.