正态分布
本节将详细介绍正态分布（Normal Distribution），这是统计学中最重要的连续概率分布之一。正态分布也称为高斯分布（Gaussian Distribution），是一种连续型概率分布，其概率密度函数呈钟形曲线，具有对称性。
This section will detail the Normal Distribution, one of the most important continuous probability distributions in statistics. Also known as the Gaussian Distribution, it is a continuous probability distribution with a bell-shaped probability density function that is symmetric.
在现实世界中，许多自然现象和社会现象都可以用正态分布来建模，如人类身高、体重、考试成绩等。通过学习本节内容，您将理解正态分布的基本特征、参数含义以及其在统计学中的重要应用。
In the real world, many natural and social phenomena can be modeled using the normal distribution, such as human height, weight, and exam scores. By studying this section, you will understand the basic characteristics of the normal distribution, the meaning of its parameters, and its important applications in statistics.
可以取无限多个值的随机变量，取任何特定值的概率为0，但可以计算在某个区间内取值的概率。
A random variable that can take an unlimited number of values, with probability 0 of taking any specific value, but the probability of values within a given range can be calculated.
正态分布由两个参数描述：均值μ（位置参数）和方差σ²（形状参数），记作X ~ N(μ, σ²)。
The normal distribution is described by two parameters: mean μ (location parameter) and variance σ² (shape parameter), denoted as X ~ N(μ, σ²).
正态分布的重要性质：约68%的数据在均值±1标准差内，95%在±2标准差内，99.7%在±3标准差内。
An important property of the normal distribution: approximately 68% of data lies within ±1 standard deviation of the mean, 95% within ±2 standard deviations, and 99.7% within ±3 standard deviations.
\[X \sim \mathrm{N}(\mu, \sigma^2)\]
其中X是正态分布的随机变量，μ是均值，σ²是方差。
Where X is a normally distributed random variable, μ is the mean, and σ² is the variance.
\[f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]