📚 Central Limit Theorem and Its Applications | 中心极限定理及其应用
The Central Limit Theorem (CLT) is a fundamental result in probability theory and statistics. It describes the distribution of the sum or average of many independent random variables, and it explains why the normal distribution is so common in real-world data.
中心极限定理（CLT）是概率论和统计学中的基本结果。它描述了许多独立随机变量之和或平均值的分布，并解释了为什么正态分布在实际数据中如此常见。
1. What is the Central Limit Theorem? | 什么是中心极限定理？
The Central Limit Theorem states that the sum of a large number of independent and identically distributed random variables, after being appropriately standardised, has an approximately normal distribution.
中心极限定理表明：大量独立同分布的随机变量之和，经过适当标准化后，近似服从正态分布。
In fact, whether the population distribution is normal or not, the sampling distribution of the sample mean X̄ becomes bell-shaped as the sample size n grows.
事实上，无论总体分布是否正态，随着样本量 n 增大，样本均值 X̄ 的抽样分布都会呈钟形。
It is the reason why many statistics in nature, business, and science follow a normal pattern in practice.
它是许多自然、商业和科学中的统计量在实践中呈现正态模式的原因。
The approximation improves as the sample size n increases, making the CLT extremely useful for statistical inference.
近似程度随着样本量 n 的增大而提高，这使得中心极限定理在统计推断中极为有用。
2. Conditions and Intuition | 适用条件与直观理解
The random variables in the sample must be independent of each other.
样本中的随机变量必须相互独立。
They must be identically distributed with a finite mean μ and a finite variance σ².
它们必须同分布，且具有有限均值 μ 和有限方差 σ²。
The sample size n should be sufficiently large; in many practical situations, n ≥ 30 is considered enough for a good approximation.
样本量 n 应足够大；在许多实际情形中，通常认为 n ≥ 30 就足以获得良好的近似。
Intuitively, the CLT works because individual fluctuations cancel out when averaged, leaving a smooth symmetric 'bell' curve.
直观上，中心极限定理之所以成立，是因为
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