大数定律和中心极限定理

The Law of Large Numbers (LLN) and the Central Limit Theorem (CLT) are two fundamental concepts in probability theory and statistics.

Law of Large Numbers (LLN):
The Law of Large Numbers states that as the size of a sample increases, the sample mean will get closer to the expected value (population mean) of the distribution. In other words, with a larger number of trials or observations, the average of the results becomes more accurate and converges to the true mean.
样本足够大，样本均值会趋向于总体均值

Central Limit Theorem (CLT):
The Central Limit Theorem states that the distribution of the sample mean of a sufficiently large number of independent, identically distributed(i.i.d) random variables approaches a normal distribution (Gaussian distribution), regardless of the original distribution of the variables. This is true as long as the sample size is large enough.当样本足够大，均值近似服从正态分布

Key Differences:

1. Purpose:

   • LLN focuses on the convergence of the sample mean to the expected value as the sample size increases.
   • CLT focuses on the shape of the distribution of the sample mean, indicating that it becomes approximately normal regardless of the original distribution.

2. Sample Size:

   • LLN requires a large sample size for the sample mean to approximate the population mean closely.
   • CLT requires a sufficiently large sample size for the distribution of the sample mean to become normal.

3. Distribution:

   • LLN does not specify the distribution of the sample mean, only its convergence to the population mean.
   • CLT specifies that the distribution of the sample mean will be approximately normal, even if the original data distribution is not normal.

In summary, while both the Law of Large Numbers and the Central Limit Theorem deal with the behavior of sample means as the sample size increases, LLN is concerned with the accuracy of the sample mean in estimating the population mean, whereas CLT is concerned with the distribution of the sample mean becoming normal.