Law of Large Numbers Simulation

Probability · Axiom Academy

EXAMPLE Law of Large Numbers Simulation Demonstrating how sample mean converges to expected value Excellent work! You've completed this example. Here's what we learned: Law of Large Numbers: As sample size increases, the sample mean converges to the expected value with probability 1. Small samples vary widely: With n=10, we saw X̄₁₀ = 0.4, which differs noticeably from E[X] = 0.5. Large samples stabilize: With n=1000, X̄₁₀₀₀ = 0.501 is very close to the true expected value. Practical importance: This theorem justifies using sample averages to estimate population means in statistics. The Law of Large Numbers is fundamental to probability and statistics, guaranteeing that empirical averages approach theoretical expectations!

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