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What is the law of large numbers?

The Law of Large Numbers is a fundamental principle in probability theory and statistics. It states that as the number of independent, identically distributed random experiments increases, the relative difference between the empirical probability of an event and its theoretical probability converges to zero.

There are two main formulations of the Law of Large Numbers:

Weak Law of Large Numbers (WLLN):

• This formulation states that the average of independent, identically distributed random variables (with finite expected value) will converge to the expected value of those random variables as the sample size increases.
• In other words, the relative deviation of the average from the expected value will tend to zero with high probability.

Strong Law of Large Numbers (SLLN):

• This formulation is stronger and asserts that with almost certain probability (probability 1), the average of independent, identically distributed random variables converges to their expected value.
• This means that the convergence occurs not just in expectation but almost surely.

The Law of Large Numbers has broad applications in various fields, including statistics, actuarial science, finance, and machine learning. It underscores the stability of statistical estimates when the sample size is large and forms the basis of many probabilistic models.