Monte Carlo Big Ideas

Concept	Representative Quantities / Methods	Purpose / Idea
Population/Truth	random variable $Y=f(\boldsymbol{X})$, mean $\mu$, variance $\sigma^2$, probability density/mass function $\varrho$, cumulative distribution function F, quantile function Q	System governed or modeled by randomness or uncertainty
Applications	quantitative finance, Bayesian inference, queueing systems, uncertainty quantification (UQ), image rendering, stochastic gradient descent	Apply Monte Carlo methodology to practical problems
Sampling	independent and identically distibuted (IID), low discrepancy (LD), quartile transformation, affine transformation, multivariate Gaussian, Markov Chain Monte Carlo (MCMC), acceptance–rejection	Generate representative points from the population
Estimation	sample mean, sample variance, histogram, kernel density estimation, empirical distribution function, empirical quantile function	Approximate population quantities using sample quantities
Efficiency	importance sampling, control variates, stratified sampling, multilevel (quasi-)Monte Carlo (ML(Q)MC)	Reduce variance/variation or computational cost for a fixed accuracy
Estimate Uncertainty	Central Limit Theorem (CLT), bootstrap, Bayesian credible intervals, tracking Fourier coefficients	Quantify the reliability of Monte Carlo estimates