The Standard Normal Distribution | 标准正态分布

📚 The Standard Normal Distribution | 标准正态分布

In Edexcel A-Level Mathematics, the standard normal distribution is the foundation for all normal probability calculations. Once you can convert any normal variable to the standard normal variable Z, you can use a single set of statistical tables to find probabilities for countless practical situations. This article explains the key ideas, methods and common exam techniques.

在 Edexcel A-Level 数学中，标准正态分布是所有正态概率计算的基础。只要能将任意正态变量转换为标准正态变量 Z，就可以使用同一套统计表解决大量实际问题中的概率计算。本文讲解核心概念、计算方法和常见考试技巧。

1. What Is the Standard Normal Distribution? | 什么是标准正态分布？

The standard normal distribution is a special normal distribution with mean μ = 0 and standard deviation σ = 1. It is written as Z ~ N(0, 1). The letter Z is used to show that the variable has already been standardised.

标准正态分布是一种特殊的正态分布，均值为 μ = 0，标准差为 σ = 1，记作 Z ~ N(0, 1)。使用字母 Z 表示该变量已经被标准化。

Its probability density function is φ(z) = 1/√(2π) × e^(−z²/2). You do not need to integrate this function in Edexcel exams because the normal table gives the required areas directly.

其概率密度函数为 φ(z) = 1/√(2π) × e^(−z²/2)。在 Edexcel 考试中无需对这个函数积分，因为正态分布表直接给出了所需面积。

The total area under the standard normal curve is exactly 1. Probabilities are therefore found as areas under this curve.

标准正态曲线下的总面积为 1。因此，概率可以通过该曲线下的面积求得。

2. Shape and Properties of the Curve | 曲线的形状与性质

The standard normal curve is bell-shaped and symmetric about the vertical line z = 0. It extends from −∞ to +∞ and never touches the horizontal axis. The maximum height occurs at z = 0 and is approximately 0.3989.

标准正态曲线呈钟形，关于竖直直线 z = 0 对称。它从 −∞ 延伸到 +∞，并且永远不会接触横轴。曲线在 z = 0 处达到最高点，高度约为 0.3989。

Because the distribution is continuous, P(Z < z) is the same as P(Z ≤ z). The endpoints do not affect the probability. Also, P(Z = a) = 0 for any single value a.

由于分布是连续的，P(Z < z) 与 P(Z ≤ z) 相同，端点不影响概率。此外，对任何单个值 a，P(Z = a) = 0。

The curve has inflection points at z = −1 and z = 1. The area between these two values is about 0.6826, which is part of the 68–95–99.7 rule discussed later.

曲线在 z = −1 和 z = 1 处有拐点。这两个值之间的面积约为 0.6826，这是后面要讨论的 68–95–99.7 规律的一部分。

3. Standardising a Normal Variable | 正态变量的标准化

If X has a normal distribution with mean μ and standard deviation σ, then the transformed variable Z = (X − μ) / σ has the standard normal distribution N(0, 1). This process is called standardising.

如果 X 服从均值为 μ、标准差为 σ 的正态分布，那么变换后的变量 Z = (X − μ) / σ 服从标准正态分布 N(0, 1)。这个过程称为标准化。

It is essential to divide by the standard deviation σ, not by the variance σ². A common mistake is to use σ² in the denominator, which gives an incorrect z-value.

必须除以标准差 σ，而不是方差 σ²。常见错误是在分母中使用 σ²，从而得到错误的 z 值。

Example: If X ~ N(50, 4²), then X has μ = 50 and σ = 4. For x = 58, the standardised value is z = (58 − 50) / 4 = 2.

例如：若 X ~ N(50, 4²)，则 μ = 50，σ = 4。当 x = 58 时，标准化值为 z = (58 − 50) / 4 = 2。

4. Using the Standard Normal Table | 使用标准正态分布表

Edexcel tables usually give the cumulative probability Φ(z) = P(Z < z) for positive z-values from 0.00 to 4.00. For example, Φ(1.25) = P(Z < 1.25) = 0.8944.

Edexcel 表格通常给出 z 从 0.00 到 4.00 的累积概率 Φ(z) = P(Z < z)。例如，Φ(1.25) = P(Z < 1.25) = 0.8944。

For negative z-values, use symmetry: Φ(−z) = 1 − Φ(z). For instance, P(Z < −1.25) = 1 − 0.8944 = 0.1056.

对于负的 z 值，利用

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）