摘要：Today, the editor brings the "Deep in the Woods, the Deer Appears (Part 7): Probability Theory and Mathematical Statistics (6)".

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“林深见鹿（七）：概率论与数理统计（6）”。

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Today, the editor brings the "Deep in the Woods, the Deer Appears (Part 7): Probability Theory and Mathematical Statistics (6)".

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思维导图

MindMapping

今天我想和大家聊聊概率论中两个非常重要且有趣的概念——大数定理和中心极限定理。这两个定理不仅是概率论课程的核心内容，更是理解随机现象背后规律的关键所在。

Today, I'd like to discuss with you two extremely important and fascinating concepts in probability theory—the Law of Large Numbers and the Central Limit Theorem. These two theorems are not only core components of probability theory courses but also the keys to understanding the underlying patterns of random phenomena.

让我们先来认识大数定理。大数定理告诉我们，当试验次数足够多时，随机事件发生的频率会逐渐稳定并趋近于其真实概率。想象一下抛硬币的情景：理论上正面朝上的概率是0.5，但如果你只抛10次，可能会出现7次正面，频率为0.7，与理论值有较大差距。但如果你坚持不懈地抛1000次、10000次，就会发现正面朝上的次数越来越接近总次数的一半。这种"频率稳定性"的现象就是大数定理的精髓所在。它在保险精算、风险评估等领域有着广泛的应用，帮助人们通过对大量数据的观察来预测随机事件的长期规律。

Let's first get to know the Law of Large Numbers. The Law of Large Numbers tells us that when the number of trials is sufficiently large, the frequency of a random event occurring will gradually stabilize and approach its true probability. Imagine the scenario of flipping a coin: theoretically, the probability of getting heads is 0.5. But if you only flip it 10 times, you might get heads 7 times, resulting in a frequency of 0.7, which shows a relatively large deviation from the theoretical value. However, if you keep flipping it 1,000 times or 10,000 times, you'll find that the number of times heads appears gets closer and closer to half of the total number of flips. This phenomenon of "frequency stability" is the essence of the Law of Large Numbers. It has a wide range of applications in fields such as insurance actuarial science and risk assessment, helping people predict the long-term patterns of random events by observing large amounts of data.

接下来了解中心极限定理，这个定理可以说更加神奇。它告诉我们，无论原始总体是什么分布，只要样本量足够大，样本均值的分布都会趋近于正态分布。举个例子，假设某高校学生的身高分布是偏态的，但如果我们多次抽取样本（每次样本量足够大），计算每次样本的平均身高，这些平均身高的分布就会呈现出经典的钟形曲线特征。这个定理为统计推断奠定了坚实基础，让我们能够利用正态分布的性质进行参数估计、假设检验等操作。

Next, let's delve into the Central Limit Theorem, which can be described as even more remarkable. It states that regardless of the distribution of the original population, as long as the sample size is sufficiently large, the distribution of the sample means will approach a normal distribution.For instance, suppose the height distribution of students in a certain university is skewed. However, if we take multiple samples (with a sufficiently large sample size each time) and calculate the average height for each sample, the distribution of these average heights will exhibit the characteristic bell-shaped curve of a classic normal distribution.This theorem lays a solid foundation for statistical inference, enabling us to perform operations such as parameter estimation and hypothesis testing by utilizing the properties of the normal distribution.

虽然这两个定理都涉及大量观测值的规律性，但它们的关注点有所不同。大数定理着重说明单个随机变量的稳定性，告诉我们频率如何趋近于概率；而中心极限定理则关注样本均值的分布形态，揭示了多个样本均值的分布规律。

Although both of these theorems deal with the regularity of a large number of observations, they focus on different aspects. The Law of Large Numbers emphasizes the stability of a single random variable, explaining how frequency approaches probability. In contrast, the Central Limit Theorem focuses on the distribution shape of sample means, revealing the distribution pattern of multiple sample means.

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参考资料：百度百科

来源：LearningYard学苑