Monte Carlo & Stochastic Simulation
Estimate pi, compute integrals, and run a random walk using random sampling techniques.
Step 1 — Estimate pi with random darts
Throw random points at a unit square. Those that land inside the quarter-circle (x²+y²≤1) approximate the area π/4. Multiply the ratio by 4 to estimate π.
N = 10000;
x = rand(1, N); y = rand(1, N);
inside = sum(x.^2 + y.^2 <= 1);
pi_est = 4 * inside / N;
printf('Pi estimate: %.4f (error: %.4f)\n', pi_est, abs(pi_est - pi))▶ Run in SimLabExpected output: Pi estimate near 3.1416, error < 0.01
Step 2 — Monte Carlo integration
Estimate ∫₀¹ √(1−x²) dx (= π/4) by sampling uniformly and averaging the integrand. This generalises to high-dimensional integrals where quadrature is impractical.
N = 5000;
x = rand(1, N);
f = sqrt(1 - x.^2);
integral_est = mean(f);
printf('Integral estimate: %.4f (pi/4 = %.4f)\n', integral_est, pi/4)▶ Run in SimLabExpected output: Integral estimate close to 0.7854 (= pi/4)
Step 3 — Random walk
A 1D random walk adds +1 or -1 at each step with equal probability. Plotting many walks shows the diffusion envelope growing as √t.
steps = 500;
walks = 5;
hold on;
for k = 1:walks
r = 2*(rand(1, steps) > 0.5) - 1;
pos = cumsum(r);
plot(1:steps, pos);
end
xlabel('Step'); ylabel('Position');
title('1D Random Walks')▶ Run in SimLabExpected output: Five random walk paths diverging from the origin
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