Finding Probabilities for Normal Distributions | 正态分布概率的计算

📚 Finding Probabilities for Normal Distributions | 正态分布概率的计算

The normal distribution is one of the most important continuous probability distributions in statistics. For a random variable X that follows a normal distribution with mean μ and standard deviation σ, written as X ~ N(μ, σ²), the total area under the probability density curve equals 1, and every probability corresponds to an area under this curve.

正态分布是统计学中最重要的连续型概率分布之一。若随机变量 X 服从均值为 μ、标准差为 σ 的正态分布，记作 X ~ N(μ, σ²)，则概率密度曲线下的总面积为 1，而每一个概率都对应曲线下的一个面积。

1. The Standard Normal Distribution | 标准正态分布

A special case of the normal distribution is the standard normal distribution, denoted Z ~ N(0, 1), which has mean 0 and standard deviation 1. Its cumulative distribution function is written as Φ(z) = P(Z ≤ z), which represents the area under the curve to the left of z.

正态分布的一个特例是标准正态分布，记作 Z ~ N(0, 1)，其均值为 0，标准差为 1。它的累积分布函数记作 Φ(z) = P(Z ≤ z)，表示曲线下 z 左侧的面积。

Every normal distribution can be converted into the standard normal distribution through standardisation. This is crucial because it means a single table can be used to find probabilities for any normal distribution.

任何正态分布都可以通过标准化转换为标准正态分布。这一点至关重要，因为它意味着我们可以用同一张表来求解任意正态分布的概率。

2. The Standardisation Formula | 标准化公式

If X ~ N(μ, σ²), then the standardised variable is calculated using the formula:

若 X ~ N(μ, σ²)，则标准化变量通过以下公式计算：

Z = (X − μ) / σ

This transformation shifts the mean to 0 and rescales the standard deviation to 1. For example, if X ~ N(50, 10²), then the value X = 65 corresponds to Z = (65 − 50) / 10 = 1.5, meaning that 65 is 1.5 standard deviations above the mean.

此变换将均值平移至 0，并将标准差缩放为 1。例如，若 X ~ N(50, 10²)，则 X = 65 对应 Z = (65 − 50) / 10 = 1.5，表示 65 高于均值 1.5 个标准差。

You must always perform this standardisation step before using the normal distribution table. The z-score tells you how many standard deviations the raw value is away from the mean, and its sign tells you whether the value is above or below the mean.

在使用正态分布表之前，必须始终完成这一标准化步骤。z 分数告诉你原始值偏离均值多少个标准差，其正负号则说明该值在均值上方还是下方。

3. Reading the Normal Distribution Table | 正态分布表的读取

In the Edexcel examination, a standard normal distribution table is provided in the formula booklet. The table gives Φ(z) = P(Z ≤ z) for non-negative values of z, with rows representing units and first decimal places, and columns representing second decimal places.

在 Edexcel 考试中，公式册中附有标准正态分布表。该表给出非负 z 值对应的 Φ(z) = P(Z ≤ z)，其中行表示整数及第一位小数，列表示第二位小数。

For example, to find Φ(1.25), locate row 1.2 and column 0.05:

例如，查找 Φ(1.25) 时，先定位行 1.2，再定位列 0.05：