Law of large numbers

The law of large numbers states that averages of repeated random observations converge toward their expected value as the number of observations grows. It explains why long-run averages stabilize without guaranteeing that short-run outcomes do.

Weak law of large numbers: A theorem stating that sample averages converge in probability to the expected value as sample size increases. This version requires the probability of a large deviation to vanish, not each sequence of averages to converge.
Expected value: The probability-weighted mean of a random variable's possible values. It is the destination of the sample averages described by the law.
Monte Carlo method: A computational method that estimates quantities by sampling random outcomes repeatedly. Its estimates stabilize as simulation runs accumulate, under suitable sampling conditions.
Central limit theorem: A theorem describing the limiting distribution of normalized sums of many random variables. It describes the shape of fluctuations around the average, while the law describes convergence of the average itself.
Strong law of large numbers: A theorem stating that sample averages converge almost surely to the expected value under specified conditions. It gives a stronger, path-by-path form of long-run convergence than the weak law.
Sample mean: The arithmetic average of observations in a finite sample. The theorem describes how this statistic behaves as the sample grows.
Monte Carlo integration: A numerical integration method that estimates an integral using averages of randomly sampled function values. The law explains why the sample average approaches the integral under appropriate conditions.
Gambler's fallacy: The mistaken belief that independent random outcomes become due to compensate for earlier outcomes. Long-run stabilization does not force a short sequence to balance itself.
Convergence in probability: A sequence of random variables converges in probability when the chance of a fixed-size error tends to zero. This is the convergence mode used in the weak law.
Random variable: A function that assigns numerical values to outcomes of a random experiment. Repeated observations are modeled as random variables whose averages can be studied.

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