Law of large numbers
Averages of many trials converge to the expected value.
The law of large numbers is a fundamental theorem in probability theory stating that the average of results obtained from a large number of independent random samples converges to the true expected value, if it exists. It guarantees stable long-term averages for random events, such as a casino's earnings over many spins of a roulette wheel, and is essential in fields including statistics, economics, and insurance.
- field
- Probability theory
- known_for
- Stating that the sample mean converges to the true mean for large independent and identically distributed samples
- first_proved_by
- Jacob Bernoulli
- named_by
- S. D. Poisson
- key_refiners
- Chebyshev, Markov, Borel, Cantelli, Kolmogorov, Khinchin
Lore & Background
This was later formalized as a law of large numbers. A special form for a binary random variable was first proved by Jacob Bernoulli, who took over 20 years to develop a rigorous proof, published in his Ars Conjectandi in 1713.
Reader's Guide
The law of large numbers is significant because it provides a mathematical foundation for the stability of long-term averages in random processes. It assures that, for independent and identically distributed random variables with a finite expected value, the sample mean converges to that expected value as the sample size grows. This principle underlies many practical applications, such as the Monte Carlo method, which relies on repeated random sampling to approximate numerical results. The law has two forms: the weak law and the strong law, the latter implying the former. However, the law has limitations: it does not apply to distributions without a finite expectation, such as the Cauchy distribution, nor does it correct selection bias in trials. Its development involved contributions from many mathematicians, including Chebyshev, Markov, Borel, Cantelli, Kolmogorov, and Khinchin.
Did You Know?
- The law of large numbers was first proved for a binary random variable by Jacob Bernoulli, who called it his 'golden theorem.'
- The law does not apply to distributions like the Cauchy distribution, which lacks an expected value.
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