📚 Applying the Central Limit Theorem to Other Distributions | 将中心极限定理应用于其他分布
In computer science, we rarely work with perfectly normal data. Arrival times of network packets, server response times, click-through rates, and random walk steps all come from non-normal distributions such as Bernoulli, Poisson, exponential, and uniform distributions. The central limit theorem (CLT) is the statistical bridge that allows us to use normal-based inference even when the original population is not normal. This article explains how to apply the CLT to these other distributions and why it is essential for algorithm analysis, simulation, and data-driven systems.
在计算机科学中，我们很少处理完全正态的数据。网络数据包的到达时间、服务器响应时间、点击率和随机游走步长都来自非正态分布，例如伯努利分布、泊松分布、指数分布和均匀分布。中心极限定理（CLT）是一座统计桥梁，它使我们即使原始总体不是正态分布，也能使用基于正态分布的推断方法。本文将解释如何将 CLT 应用于其他分布，以及它在算法分析、模拟和数据驱动系统中的重要性。
1. Why the CLT Matters in Computer Science | 中心极限定理为何在计算机科学中重要
Many computer science problems involve sums or averages of independent random events: the total number of successful requests in a benchmarking run, the average processing time over thousands of jobs, or the proportion of users who click an advertisement. Even when individual events are skewed or discrete, the average of many independent observations tends to follow a normal distribution under mild conditions. This makes the CLT a practical tool for estimating error bars, constructing confidence intervals, and deciding whether observed differences are statistically significant.
许多计算机科学问题涉及独立随机事件的和或平均值：基准测试运行中成功请求的总数、数千个作业的平均处理时间，或点击广告的用户比例。即使单个事件是偏斜或离散的，在温和条件下，大量独立观测值的平均值往往服从正态分布。这使得 CLT 成为估计误差范围、构建置信区间以及判断观察到的差异是否具有统计显著性的实用工具。
For example, a simulation may generate a large number of random samples from a skewed distribution. Instead of needing the exact sampling distribution of the mean, we can use the CLT to approximate it with a normal distribution, provided the sample size is large enough. This approximation is widely used in Monte Carlo integration, randomised algorithms, and performance evaluation of computer systems.
例如，模拟可能从偏斜分布中生成大量随机样本。我们并不需要均值的精确抽样分布，只要样本量足够大，就可以利用 CLT 用正态分布来近似它。这种近似广泛用于蒙特卡洛积分、随机化算法以及计算机系统的性能评估。
2. The Core Statement of the CLT | 中心极限定理的核心表述
ributed (i.i.d.) random variables with mean μ and finite variance σ². The sample mean is X̄ = (X₁ + X₂ + ... + Xₙ) / n. The CLT states that as n increases, the distribution
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