摘要：大数定律说明了随机变量的样本平均随着样本量的增大趋于其期望值。具体来说，当样本量趋于无穷大时，样本平均几乎肯定地收敛到真实的均值。

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思维导图

Mind mapping

大数定律（LLN）

The Law of Large numbers (LLN)

大数定律说明了随机变量的样本平均随着样本量的增大趋于其期望值。具体来说，当样本量趋于无穷大时，样本平均几乎肯定地收敛到真实的均值。

The Law of Large Numbers can be understood as follows: If you repeat the same experiment many times, such as flipping a coin, the average result you get (like the frequency of heads) will increasingly approach the theoretical expected value (in the case of a coin flip, the proportion of heads will get closer to 50%). In simple terms, it's the idea that "the average result of a large number of repeated experiments will tend to stabilize."

中心极限定理（CLT）

The Central Limit Theorem (CLT)

中心极限定理说明了当独立随机变量足够多时，它们的和（经过标准化处理）的分布会趋近于正态分布。也就是说，无论原始随机变量的分布如何，它们的和的分布将呈现对称的钟形曲线。

The Central Limit Theorem states that: When you have a large number of independent random variables and you sum them up, the distribution of that total, after appropriate standardization (that is, subtracting the mean and dividing by the standard deviation), will become more and more like a normal distribution. This means that no matter what the original distribution of these random variables is, once you add them together, the resulting distribution will tend to form a symmetrical, bell-shaped curve.

两者关系

Relationship Between the Two

大数定律告诉我们样本平均会收敛到真实的均值。中心极限定理告诉我们，随着样本量的增大，样本平均的分布会趋近于正态分布。

The Law of Large Numbers tells us that as the number of experiments increases, we can estimate a constant probability or average more accurately.The Central Limit Theorem tells us that when we consider many such estimates, the distribution of these estimated values will exhibit a specific shape (the normal distribution), which allows us to make more precise predictions and analyses of experimental results.

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参考资料：谷歌翻译、百度、B站

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