STRONG LAW OF LARGE NUMBERS Meaning and
Definition
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The strong law of large numbers is a fundamental principle in probability theory and statistics that describes the behavior of the average of a large number of independent, identically distributed random variables. It states that as the sample size increases indefinitely, the sample average converges to the population mean almost surely.
In more precise terms, let X₁, X₂, X₃, ... be a sequence of independent and identically distributed random variables with a common distribution and finite mean μ. Then, according to the strong law of large numbers, the sample average (X₁ + X₂ + X₃ + ... + Xₙ) / n converges to the population mean μ as n approaches infinity, with probability 1.
This law emphasizes that in the context of a large number of independent observations, the average of those observations becomes increasingly precise and reliable in estimating the true mean of the underlying distribution.
The strong law of large numbers is a powerful and indispensable concept in statistics and has important implications in various fields such as financial analysis, quality control, and scientific studies. It provides the theoretical foundation for the concept of statistical consistency, as it guarantees that the sample average will eventually, with almost certain probability, approach the true mean as more data is gathered.
