在统计学中，连续随机变量是可以取无限多个值的随机变量。与离散随机变量不同，连续随机变量取任何特定值的概率为0，我们只能计算它在某个区间内取值的概率。
In statistics, a continuous random variable is a random variable that can take an unlimited number of values. Unlike discrete random variables, the probability that a continuous random variable takes any specific value is 0; we can only calculate the probability that it takes values within a certain interval.
对于连续随机变量X，有以下重要性质：
For a continuous random variable X, the following important properties hold:
1. \( P(X = a) = 0 \) 对于任何实数a
1. \( P(X = a) = 0 \) for any real number a
2. \( P(a \leq X \leq b) = P(a < X < b) \) 因为单点概率为0
2. \( P(a \leq X \leq b) = P(a < X < b) \) because single point probability is 0
正态分布是最常见的连续概率分布之一，其概率密度函数（PDF）呈钟形曲线。正态分布由两个参数完全描述：均值μ和方差σ²，记作X ~ N(μ, σ²)。
The normal distribution is one of the most common continuous probability distributions, with a bell-shaped probability density function (PDF). It is completely described by two parameters: the mean μ and the variance σ², denoted as X ~ N(μ, σ²).
正态分布的概率密度函数为：
The probability density function of the normal distribution is:
\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]
正态分布的主要特征包括：
The main characteristics of the normal distribution include:
正态分布的一个重要性质是经验法则，它描述了数据在均值附近的分布情况：
An important property of the normal distribution is the empirical rule, which describes how data is distributed around the mean:
• 约68%的数据位于均值±1个标准差范围内：\( P(\mu - \sigma \leq X \leq \mu + \sigma) \approx 0.68 \)
• Approximately 68% of the data lies within ±1 standard deviation of the mean: \( P(\mu - \sigma \leq X \leq \mu + \sigma) \approx 0.68 \)
• 约95%的数据位于均值±2个标准差范围内：\( P(\mu - 2\sigma \leq X \leq \mu + 2\sigma) \approx 0.95 \)
• Approximately 95% of the data lies within ±2 standard deviations of the mean: \( P(\mu - 2\sigma \leq X \leq \mu + 2\sigma) \approx 0.95 \)
• 约99.7%的数据位于均值±3个标准差范围内：\( P(\mu - 3\sigma \leq X \leq \mu + 3\sigma) \approx 0.997 \)
• Approximately 99.7% of the data lies within ±3 standard deviations of the mean: \( P(\mu - 3\sigma \leq X \leq \mu + 3\sigma) \approx 0.997 \)
经验法则是一个近似规则，在实际应用中非常有用，可以快速评估数据的分布情况。
The empirical rule is an approximate rule that is very useful in practical applications, allowing for quick assessment of data distribution.
一家工厂生产的金属销直径服从正态分布N(10, 0.04)，其中单位为毫米。计算以下概率：
The diameters of metal pins produced by a factory follow a normal distribution N(10, 0.04) in millimeters. Calculate the following probabilities:
解 / Solution:
已知均值μ = 10，方差σ² = 0.04，因此标准差σ = 0.2
Given mean μ = 10, variance σ² = 0.04, so standard deviation σ = 0.2
1. 9.6 = 10 - 2(0.2) = μ - 2σ，10.4 = 10 + 2(0.2) = μ + 2σ
根据68-95-99.7法则，约95%的数据在μ±2σ范围内
因此，P(9.6 ≤ X ≤ 10.4) ≈ 0.95
1. 9.6 = 10 - 2(0.2) = μ - 2σ, 10.4 = 10 + 2(0.2) = μ + 2σ
According to the 68-95-99.7 rule, approximately 95% of data lies within μ±2σ
Therefore, P(9.6 ≤ X ≤ 10.4) ≈ 0.95
2. 10.3 = 10 + 1.5(0.2) = μ + 1.5σ
由于正态分布的对称性，我们知道：
• P(μ - σ ≤ X ≤ μ + σ) ≈ 0.68 → P(X ≤ μ + σ) ≈ 0.84
• P(μ - 2σ ≤ X ≤ μ + 2σ) ≈ 0.95 → P(X ≤ μ + 2σ) ≈ 0.975
1.5σ位于1σ和2σ之间，通过线性近似估算：
P(X > 10.3) = P(X > μ + 1.5σ) ≈ 0.067
2. 10.3 = 10 + 1.5(0.2) = μ + 1.5σ
Due to the symmetry of the normal distribution, we know:
• P(μ - σ ≤ X ≤ μ + σ) ≈ 0.68 → P(X ≤ μ + σ) ≈ 0.84
• P(μ - 2σ ≤ X ≤ μ + 2σ) ≈ 0.95 → P(X ≤ μ + 2σ) ≈ 0.975
1.5σ is between 1σ and 2σ, using linear approximation:
P(X > 10.3) = P(X > μ + 1.5σ) ≈ 0.067
上述例子中使用的是经验法则的近似值。在实际应用中，我们需要使用标准正态分布表或计算器来获取更精确的概率值。
The above example uses approximate values from the empirical rule. In practical applications, we need to use standard normal distribution tables or calculators to obtain more precise probability values.
正态分布在统计学中占据中心地位，这主要归功于以下几个原因：
The normal distribution occupies a central position in statistics, mainly due to the following reasons: