Random Numbers in Statistics: Sampling, Bootstrap, and Monte Carlo

Randomness underpins modern statistics—from unbiased sampling to simulation‑based inference. Here’s how it’s used and how to use it well.

Core Applications

• Random sampling: Select representative subsets to estimate population parameters without systematic bias.
• Bootstrap: Resample observed data with replacement to estimate uncertainty (CIs, standard errors) when analytic formulas are hard.
• Monte Carlo simulation: Approximate complex probabilities and expectations by repeated random draws.

Practical Tips

• PRNG choice: Use high‑quality PRNGs designed for scientific computing (e.g., PCG/MT variants) rather than cryptographic RNGs.
• Seeding: Fixed seeds for reproducibility in research; document seeds in methods sections and notebooks.
• Distribution transforms: Map uniform[0,1) to target distributions via inverse‑CDF, Box–Muller, or library routines.
• Diagnostics: Validate with known results on small problems, then scale up.

Mini‑Examples

• Bootstrap CI: Draw 10,000 resamples of your statistic (mean, median), compute percentiles for a 95% CI.
• Pi via Monte Carlo: Sample points in [0,1]^2; share of points within the unit quarter‑circle × 4 approximates π.
• Sampling bias check: Compare random sample means vs full dataset to ensure no drift.

Common Gotchas

• Correlated draws: Don’t reuse the same draws across comparisons unless intended; isolate streams.
• Small samples: Monte Carlo error can dominate; increase trials and report uncertainty.
• State leakage: Reset or manage PRNG state between experiments to avoid cross‑contamination.

FAQs

Do I need cryptographic RNGs? No. For statistics and simulations, high‑quality PRNGs are ideal—faster and designed for this purpose. Use CSPRNGs only for security contexts.