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.
也称为高斯分布，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。由均值μ和方差σ²完全描述，记作X ~ N(μ, σ²)。
Also known as Gaussian distribution, it is a continuous probability distribution with a bell-shaped probability density function that is symmetric. It is completely described by mean μ and variance σ², denoted as X ~ N(μ, σ²).
描述连续随机变量在某一点附近取值概率的函数。对于正态分布，其概率密度函数在均值处达到最大值。
A function that describes the probability of a continuous random variable taking values near a specific point. For the normal distribution, the PDF reaches its maximum at the mean.
描述正态分布数据在均值附近分布的近似规则，指出约68%、95%和99.7%的数据分别位于均值±1、±2和±3个标准差范围内。
An approximate rule describing the distribution of normal data around the mean, stating that approximately 68%, 95%, and 99.7% of data lie within ±1, ±2, and ±3 standard deviations of the mean, respectively.
\[ X \sim \mathrm{N}(\mu, \sigma^2) \]
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.
\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]
其中σ是标准差，π是圆周率，e是自然对数的底数。
Where σ is the standard deviation, π is pi, and e is the base of natural logarithm.
• \( P(\mu - \sigma \leq X \leq \mu + \sigma) \approx 0.68 \)
• \( P(\mu - 2\sigma \leq X \leq \mu + 2\sigma) \approx 0.95 \)
• \( P(\mu - 3\sigma \leq X \leq \mu + 3\sigma) \approx 0.997 \)
1. 理解正态分布的参数意义：均值μ决定分布的位置，方差σ²决定分布的形状（分散程度）。
1. Understand the meaning of normal distribution parameters: mean μ determines the position of the distribution, and variance σ² determines the shape (spread) of the distribution.
2. 熟练掌握经验法则，可以快速估算概率，这在实际应用中非常有用。
2. Master the empirical rule to quickly estimate probabilities, which is very useful in practical applications.
3. 注意区分离散分布和连续分布的概念差异，特别是在计算概率时的不同方法。
3. Pay attention to the conceptual differences between discrete and continuous distributions, especially the different methods when calculating probabilities.
4. 正态分布是许多统计方法的基础，后续学习的标准正态分布、中心极限定理等都与之密切相关。
4. The normal distribution is the foundation of many statistical methods, and subsequent learning of standard normal distribution, central limit theorem, etc., are closely related to it.