Sentences with STRONG-LAW-OF-LARGE-NUMBERS

"The strong law of large numbers states that if a sequence of independent and identically distributed random variables has a finite mean, then the sample mean converges almost surely to the true mean."

"According to the strong law of large numbers, as the number of trials increases, the empirical probability converges to the theoretical probability."

"The strong law of large numbers is a fundamental result in probability theory."

"In gambling, the strong law of large numbers suggests that as the number of bets increases, the actual ratio of wins to total bets will converge to the expected ratio."

"The strong law of large numbers has important applications in statistics and econometrics."

"Under the strong law of large numbers, the relative frequency of an event approaches the probability of that event as the number of observations increases."

"The strong law of large numbers is sometimes referred to as Kolmogorov's strong law."

"The proof of the strong law of large numbers relies on the Borel-Cantelli lemma."

"The strong law of large numbers is a powerful tool for analyzing the behavior of random variables."

"The strong law of large numbers guarantees that the sample average will converge to the population average with probability 1."

"The strong law of large numbers is closely related to the central limit theorem."

"The strong law of large numbers can be applied to various fields including finance, biology, and physics."

"The strong law of large numbers provides a theoretical foundation for statistical inference."

"The strong law of large numbers is based on the idea that, as the sample size increases, the sample mean becomes a better estimate of the population mean."

"The strong law of large numbers is one of the key principles in probability theory."

"The strong law of large numbers ensures that the observed frequencies of outcomes will converge to the true probabilities as the number of trials increases."

"The strong law of large numbers is a fundamental concept in modern mathematics."

"The strong law of large numbers can be used to analyze the long-term performance of investment portfolios."

"The strong law of large numbers is a direct consequence of the law of large numbers."

"The strong law of large numbers has practical implications in risk management and decision-making."