Monte Carlo Simulation Cheat Sheet

Updated 2026-05-15

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Monte Carlo simulation is a computational technique that uses random sampling to estimate numerical results for problems that are analytically intractable or too complex for closed-form solutions. Named after the Monte Carlo casino in Monaco, the method transforms deterministic mathematical problems into probabilistic experiments by generating thousands to millions of random scenarios and aggregating their outcomes. At its core, Monte Carlo relies on the Law of Large Numbers—as sample size increases, the sample mean converges to the expected value—making it invaluable across finance, physics, engineering, and data science. The key insight: rather than solving an equation directly, you simulate the randomness inherent in the system and let statistics reveal the answer.

What This Cheat Sheet Covers

This topic spans 10 focused tables and 60 indexed concepts, 53 flashcards. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.

Table 1: Random Sampling FundamentalsTable 2: Pseudo-Random Number GenerationTable 3: Monte Carlo IntegrationTable 4: Variance Reduction TechniquesTable 5: Financial Monte Carlo ApplicationsTable 6: Statistical Testing ApplicationsTable 7: Markov Chain Monte Carlo (MCMC)Table 8: Convergence and DiagnosticsTable 9: Advanced Sampling TechniquesTable 10: NumPy/SciPy Implementation

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Table 1: Random Sampling Fundamentals

Everything in Monte Carlo starts with drawing random samples from a distribution, and these are the foundational recipes for doing it. Inverse transform and acceptance-rejection let you sample from almost any distribution, while crude, hit-or-miss, and sample-mean estimators show the different ways those samples turn into an answer — with importance sampling hinting at how to make each sample count for more.

Technique	Example	Description
Inverse transform sampling	`u = rng.random()` `x = F_inv(u)`	• Generates samples from any distribution by applying the inverse CDF $F^{-1}$ to uniform random numbers • works because $F(X) \sim \text{Unif}(0,1)$ for any random variable $X$.
Acceptance-rejection sampling	`x = q.rvs()` `u = rng.random()` `if u <= f(x)/(M*q(x)): accept`	• Samples from target density $f(x)$ using proposal $q(x)$ with envelope $M \cdot q(x) \geq f(x)$ • accepts with probability $\frac{f(x)}{M \cdot q(x)}$.
Crude Monte Carlo	`samples = [f(x) for x in X]` `estimate = mean(samples)`	• Estimates $\mathbb{E}[f(X)]$ by averaging $f(X_i)$ over $N$ independent samples • simplest Monte Carlo estimator with convergence rate $O(N^{-1/2})$.
Hit-or-miss integration	`hits = sum(y < f(x))` `area = (b-a) * h * hits/N`	• Estimates integral by counting random points falling under curve $f(x)$ in bounding box • intuitive but higher variance than sample-mean method
Sample mean Monte Carlo	`I = (b-a) * mean([f(x_i)]`	• Approximates $\int_a^b f(x)dx$ as $(b-a) \cdot \frac{1}{N}\sum f(X_i)$ where $X_i \sim \text{Unif}(a,b)$ • lower variance than hit-or-miss
Importance sampling	`w = f(x) / q(x)` `estimate = mean(w * h(x))`	• Samples from proposal $q(x)$ and reweights by likelihood ratio $w(x) = f(x)/q(x)$ to estimate $\mathbb{E}_f[h(X)]$ • reduces variance when $q$ concentrates where $\lvert h(x)f(x)\rvert$ is large

Table 2: Pseudo-Random Number Generation

The randomness a simulation depends on actually comes from a deterministic generator, and in modern NumPy that means default_rng and the PCG64 bit generator rather than the legacy np.random.seed. Seeding gives you reproducible runs for debugging, and once you have a generator the same object draws uniform, normal, exponential, and correlated multivariate samples.

Method	Example	Description
numpy.random.default_rng	`rng = np.random.default_rng(seed=42)`	• Creates a new Generator instance using PCG64 bit generator • recommended over legacy `np.random.seed()` for reproducible isolated RNG state

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Table 3: Monte Carlo Integration

One of the oldest and most intuitive uses of the method is approximating integrals that have no closed form — average the function at random points and scale by the domain. The famous pi estimate is the textbook demonstration, and once you move to many dimensions Monte Carlo becomes competitive precisely where grid methods collapse under the curse of dimensionality.

Method	Example	Description
Basic MC integration	`I = (b-a) * np.mean(f(X))`	Estimates $\int_a^b f(x)dx$ by averaging function evaluations at random points $X_i \sim \text{Unif}(a,b)$.

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Table 4: Variance Reduction Techniques

Because Monte Carlo error shrinks only as $1/\sqrt{N}$, brute-forcing more samples gets expensive fast — these techniques cut the variance instead, squeezing a far better estimate out of the same budget. Antithetic and control variates exploit correlation, stratification guarantees coverage, and common random numbers sharpen comparisons between two systems.

Technique	Example	Description
Antithetic variates	`Z1 = rng.normal(N)` `Z2 = -Z1` `Y = 0.5*(f(Z1)+f(Z2))`	• Pairs samples with negatively correlated counterparts (e.g., $Z$ and $-Z$) • when $f$ is monotonic, $\text{Var}(Y) < \text{Var}(f(Z))$ dramatically

The rest of this section is part of the full cheat sheet

Table 1: Random Sampling Fundamentals

Technique	Example	Description
Inverse transform sampling	`u = rng.random()` `x = F_inv(u)`	• Generates samples from any distribution by applying the inverse CDF $F^{-1}$ to uniform random numbers • works because $F(X) \sim \text{Unif}(0,1)$ for any random variable $X$.
Acceptance-rejection sampling	`x = q.rvs()` `u = rng.random()` `if u <= f(x)/(M*q(x)): accept`	• Samples from target density $f(x)$ using proposal $q(x)$ with envelope $M \cdot q(x) \geq f(x)$ • accepts with probability $\frac{f(x)}{M \cdot q(x)}$.
Crude Monte Carlo	`samples = [f(x) for x in X]` `estimate = mean(samples)`	• Estimates $\mathbb{E}[f(X)]$ by averaging $f(X_i)$ over $N$ independent samples • simplest Monte Carlo estimator with convergence rate $O(N^{-1/2})$.
Hit-or-miss integration	`hits = sum(y < f(x))` `area = (b-a) * h * hits/N`	• Estimates integral by counting random points falling under curve $f(x)$ in bounding box • intuitive but higher variance than sample-mean method
Sample mean Monte Carlo	`I = (b-a) * mean([f(x_i)]`	• Approximates $\int_a^b f(x)dx$ as $(b-a) \cdot \frac{1}{N}\sum f(X_i)$ where $X_i \sim \text{Unif}(a,b)$ • lower variance than hit-or-miss
Importance sampling	`w = f(x) / q(x)` `estimate = mean(w * h(x))`	• Samples from proposal $q(x)$ and reweights by likelihood ratio $w(x) = f(x)/q(x)$ to estimate $\mathbb{E}_f[h(X)]$ • reduces variance when $q$ concentrates where $\lvert h(x)f(x)\rvert$ is large

Table 2: Pseudo-Random Number Generation

Method	Example	Description
numpy.random.default_rng	`rng = np.random.default_rng(seed=42)`	• Creates a new Generator instance using PCG64 bit generator • recommended over legacy `np.random.seed()` for reproducible isolated RNG state

Table 3: Monte Carlo Integration

Method	Example	Description
Basic MC integration	`I = (b-a) * np.mean(f(X))`	Estimates $\int_a^b f(x)dx$ by averaging function evaluations at random points $X_i \sim \text{Unif}(a,b)$.

Table 4: Variance Reduction Techniques

Technique	Example	Description
Antithetic variates	`Z1 = rng.normal(N)` `Z2 = -Z1` `Y = 0.5*(f(Z1)+f(Z2))`	• Pairs samples with negatively correlated counterparts (e.g., $Z$ and $-Z$) • when $f$ is monotonic, $\text{Var}(Y) < \text{Var}(f(Z))$ dramatically