Definition

1. Noun:
   • A fundamental theorem of probability and statistics: The law of large numbers states that as the size of a random sample from a population increases, the sample's average (or other statistic) will tend to get closer and closer to the expected value or true average of the entire population.

Usage

• The law of large numbers is a core concept used to explain why performing more trials in an experiment leads to more stable and predictable average results.
• It is crucial for understanding reliability in fields like insurance, gambling, and scientific sampling.

Examples

Advanced Usage

• Strong Law vs. Weak Law: In advanced probability theory, the law of large numbers has more precise formulations. The states that convergence happens with absolute certainty (probability 1), while the states that convergence happens in probability.
  • The Strong Law of Large Numbers provides a more robust guarantee of convergence than the Weak Law.

Variants and Related Words

• LLN: A common abbreviation for "law of large numbers."
• Central Limit Theorem: A related but distinct statistical theorem describing the distribution of the sample mean, often studied alongside the law of large numbers.

Synonyms

• Bernoulli's theorem (specifically for the historical, simpler case).

Related Concepts

• Expected Value: The theoretical average the sample mean approaches, according to the law of large numbers.
• Sample Mean: The average of the observed data, which converges to the population mean.
• Probability Limit: The value that a sequence of random variables converges to.

Noun

1. (statistics) law stating that a large number of items taken at random from a population will (on the average) have the population statistics