How Monte Carlo Works

As we draw more samples, the histogram starts to resemble a uniform distribution. The mean is settling closer to 0.5.

The running mean is settling down and approaching 0.5. Fluctuations are smaller, but the estimate still wanders around the true value.

As we throw more darts, the estimate improves. The area of a quarter-circle with radius 1 is π/4, so multiplying the ratio by 4 gives us π. More darts = better estimate.

With few samples, each random x-value contributes a noisy estimate of the integral. The average of f(xᵢ) values is a rough approximation of the area under sin(πx).

The proposal distribution closely matches the target peak. Importance sampling concentrates samples where the target is large, reducing variance compared to uniform sampling.

Good balance! The chain accepts enough proposals to move efficiently while rejecting enough to avoid wandering too far. In 2D, acceptance rates around 40–70% give efficient exploration.