📚 pdfjoiner_(4)-216: Solving Normal Distribution with Unknown Parameters | 正态分布未知参数求解
In this article, based on the Edexcel A-Level Maths material found in pdfjoiner_(4)-216, we explore how to solve problems where the normal distribution has an unknown mean μ and/or an unknown standard deviation σ. This is a classic exam topic that tests your ability to set up simultaneous equations using the standard normal Z‑scores and inverse normal tables. You will learn to handle questions such as ‘X ~ N(μ, σ²), given P(X < a) = p and P(X > b) = q, find μ and σ’.
本文基于 Edexcel A-Level 数学资料集 pdfjoiner_(4)-216 页的内容，重点讲解如何求解正态分布中未知均值 μ 或标准差 σ 的问题。这类题目是考试的经典考点，考察你使用标准正态 Z 值和逆正态表格建立方程组的能力。你将学会处理诸如“X ~ N(μ, σ²)，已知 P(X < a) = p 且 P(X > b) = q，求 μ 和 σ”这样的问题。
1. The Normal Distribution Refresher | 正态分布回顾
Any normal random variable X with mean μ and variance σ² is written as X ~ N(μ, σ²). The probability density function is symmetric about μ, and the total area under the curve is 1. For any given value x, the probability P(X < x) is found by standardising to the Z‑score.
任何具有均值 μ 和方差 σ² 的正态随机变量 X 可记作 X ~ N(μ, σ²)。其概率密度函数关于 μ 对称，曲线下总面积为 1。对于任意给定值 x，概率 P(X < x) 需通过标准化转换为 Z 值进行求解。
2. Standardising to the Z‑score | 标准化与 Z 值
To convert an observation x into a standard normal variable Z ~ N(0,1), we use the formula Z = (X – μ) / σ. This transformation allows us to use the standard normal distribution table or calculator to find probabilities. The inverse process, Z = (x – μ) / σ, is also key when we know the probability and want to find the corresponding x, μ or σ.
要将观测值 x 转换为标准正态变量 Z ~ N(0,1)，我们使用公式 Z = (X – μ) / σ。通过这一变换，我们就能利用标准正态分布表或计算器查出概率。反过来，当已知概率而需要求对应的 x、μ 或 σ 时，逆用该公式也十分关键。
3. Using the Inverse Normal Function | 逆正态函数的使用
When you are given P(X < x) = p, you can find the Z‑score corresponding to the cumulative probability p using the inverse normal function, often denoted as Φ⁻¹(p). For example, if P(Z < z) = 0.9, then z = 1.2816 (to 4 decimal places). This z is used in the equation (x - μ) / σ = z to link μ and σ.
当给出 P(X < x) = p 时，你可以使用逆正态函数（常记作 Φ⁻¹(p)）求出累积概率 p 对应的 Z 值。例如，若 P(Z < z) = 0.9，则 z = 1.2816（保留四位小数）。将这个 z 代入方程 (x - μ) / σ = z，就能把 μ 和 σ 联系起来。
4. Problem Setup with Unknown Mean and Standard Deviation | 含未知均值和标准差的问题构建
Many Edexcel exam questions give two probability statements about a normal variable, such as P(X < 50) = 0.3 and P(X > 80) = 0.2. By converting these into Z‑score equations, you obtain two equations: (50 – μ)/σ = z₁ and (80 – μ)/σ = z₂, where z₁ and z₂ are known from the inverse normal table. These simultaneous equations can then be solved for μ and σ.
许多 Edexcel 考题会给出关于正态变量的两个概率陈述，例如 P(X < 50) = 0.3 和 P(X > 80) = 0.2。将这些条件转换为 Z 值方程，就能得到两个式子：(50 – μ)/σ = z₁ 以及 (80 – μ)/σ = z₂，其中 z₁ 和 z₂ 可通过逆正态表查出。随后解此方程组即可求出 μ 和 σ。
5. Converting Upper Tail Probabilities | 上尾概率的转换
If the problem gives an upper tail probability, such as P(X > 60) = 0.1, we first convert it to a left‑tail probability: P(X < 60) = 0.9. Then the corresponding Z‑score is Φ⁻¹(0.9). Always be careful with the direction of the inequality to avoid sign errors.
若题目给出的是上尾概率，如 P(X > 60) = 0.1，我们应首先将其转化为左尾概率：P(X < 60) = 0.9，然后对应的 Z 值就是 Φ⁻¹(0.9)。务必注意不等号的方向，以免正负号出错。
6. Worked Example 1 – Both Probabilities Given Directly | 示例 1——直接给定两个概率
Problem: The random variable X ~ N(μ, σ²). It is known that P(X < 45) = 0.25 and P(X > 65) = 0.15. Find μ and σ.
题目：随机变量 X ~ N(μ, σ²)。已知 P(X < 45) = 0.25 且 P(X > 65) = 0.15，求 μ 和 σ。
Step 1: For P(X < 45) = 0.25, the Z‑score z₁ = Φ⁻¹(0.25) = -0.6745 (since 0.25 < 0.5, z is negative). Equation: (45 - μ)/σ = -0.6745.
步骤 1：对于 P(X < 45) = 0.25，Z 值 z₁ = Φ⁻¹(0.25) = -0.6745（因 0.25 < 0.5，z 为负）。方程：(45 - μ)/σ = -0.6745。
Step 2: P(X > 65) = 0.15 means P(X < 65) = 0.85, z₂ = Φ⁻¹(0.85) = 1.0364. Equation: (65 - μ)/σ = 1.0364.
步骤 2：P(X > 65) = 0.15 即 P(X < 65) = 0.85，z₂ = Φ⁻¹(0.85) = 1.0364。方程：(65 - μ)/σ = 1.0364。
Step 3: Solve simultaneously. From equation 1, μ = 45 + 0.6745σ. Substitute into equation 2: (65 – (45 + 0.6745σ))/σ = 1.0364 → (20 – 0.6745σ)/σ = 1.0364 → 20/σ – 0.6745 = 1.0364 → 20/σ = 1.7109 → σ = 20/1.7109 ≈ 11.69. Then μ = 45 + 0.6745 × 11.69 ≈ 52.88. Thus μ ≈ 52.9, σ ≈ 11.7.
步骤 3：联立求解。由方程 1 得 μ = 45 + 0.6745σ。代入方程 2：(65 – (45 + 0.6745σ))/σ = 1.0364 → (20 – 0.6745σ)/σ = 1.0364 → 20/σ – 0.6745 = 1.0364 → 20/σ = 1.7109 → σ = 20/1.7109 ≈ 11.69。然后 μ = 45 + 0.6745 × 11.69 ≈ 52.88。因此 μ ≈ 52.9，σ ≈ 11.7。
7. Worked Example 2 – Percentage Points in Context | 示例 2——结合背景的百分位数
The lifetime of a battery is normally distributed. Only 5% of batteries last less than 100 hours, and 10% last more than 200 hours. Find the mean and standard deviation of the lifetime.
某电池寿命服从正态分布。只有 5% 的电池寿命低于 100 小时，10% 的电池寿命超过 200 小时。求寿命的均值和标准差。
Given P(X < 100) = 0.05, z₁ = Φ⁻¹(0.05) = -1.6449. For the upper tail, P(X > 200) = 0.10 implies P(X < 200) = 0.90, z₂ = Φ⁻¹(0.90) = 1.2816.
由 P(X < 100) = 0.05，得 z₁ = Φ⁻¹(0.05) = -1.6449。对于上尾，P(X > 200) = 0.10 意味着 P(X < 200) = 0.90，z₂ = Φ⁻¹(0.90) = 1.2816。
Equations: (100 – μ)/σ = -1.6449 → μ = 100 + 1.6449σ. (200 – μ)/σ = 1.2816 → μ = 200 – 1.2816σ. Equating: 100 + 1.6449σ = 200 – 1.2816σ → 2.9265σ = 100 → σ ≈ 34.17, μ ≈ 100 + 1.6449 × 34.17 ≈ 156.2. So mean ≈ 156 hours, SD ≈ 34.2 hours.
建立方程：(100 – μ)/σ = -1.6449 → μ = 100 + 1.6449σ。(200 – μ)/σ = 1.2816 → μ = 200 – 1.2816σ。令两式相等：100 + 1.6449σ = 200 – 1.2816σ → 2.9265σ = 100 → σ ≈ 34.17，μ ≈ 100 + 1.6449 × 34.17 ≈ 156.2。故均值约为 156 小时，标准差约 34.2 小时。
8. Common Pitfalls and How to Avoid Them | 常见错误与规避方法
Sign of Z: Always check whether the cumulative probability is less than or greater than 0.5 to determine the sign of the Z‑score. For example, Φ⁻¹(0.1) is negative. Tail conversion: Convert P(X > k) = p to P(X < k) = 1 - p before looking up the Z‑value. Rounding: Use at least 4 decimal places for Z‑scores to ensure accuracy in final answers.
Z 值的符号：始终检查累积概率是小于还是大于 0.5，以确定 Z 值的正负。例如 Φ⁻¹(0.1) 为负值。尾部转换：在查 Z 值之前，务必将 P(X > k) = p 转换为 P(X < k) = 1 - p。取整精度：Z 值至少保留四位小数，以确保最终答案的准确性。
9. Using the Standard Normal Table in Reverse | 标准正态表的反向使用
The Edexcel formula booklet provides a table of percentage points for the normal distribution. You can use the table giving values of z such that P(Z > z) = p for some p. Remember that P(Z > z) = p is equivalent to P(Z < -z) = p, helping you match the correct sign. Practice looking up both the main body of the standard normal table and the percentage points table.
Edexcel 公式册提供了正态分布的百分位数表。你可以利用给出 P(Z > z) = p 的表格。切记 P(Z > z) = p 等价于 P(Z < -z) = p，从而帮助你匹配正确的符号。建议练习同时查阅标准正态表主体和百分位数表。
10. Applying to Real Exam Questions | 真题实战应用
In a typical Edexcel question, you might be told that the weights of a product are normally distributed, and that 20% weigh less than 505g while 15% weigh more than 515g. You will be required to find μ and σ. The method remains identical: set up simultaneous equations and solve. Show all steps clearly, and always state the distribution of X: X ~ N(μ, σ²).
在典型的 Edexcel 考题中，可能会告诉你某产品的重量服从正态分布，20% 的重量低于 505g，15% 超过 515g，要求你求出 μ 和 σ。解题方法完全一致：建立方程组并求解。需要清晰展示所有步骤，并始终写明 X 的分布：X ~ N(μ, σ²)。
11. Check Your Understanding – Quick Exercise | 自我检测——快速练习
Suppose X ~ N(μ, σ²) and P(X < 12) = 0.3, P(X > 20) = 0.4. Find μ and σ. (Hint: Φ⁻¹(0.3) = -0.5244, Φ⁻¹(0.6) = 0.2533). Solve to confirm μ ≈ 14.5, σ ≈ 5.76. Try it yourself before checking.
假设 X ~ N(μ, σ²)，且 P(X < 12) = 0.3，P(X > 20) = 0.4。求 μ 和 σ。（提示：Φ⁻¹(0.3) = -0.5244，Φ⁻¹(0.6) = 0.2533）。动手算算看，答案应为 μ ≈ 14.5，σ ≈ 5.76。
12. Summary and Key Takeaways | 总结与核心要点
When solving for unknown parameters in a normal distribution, always standardise to Z‑scores, carefully handle tail probabilities, and set up two equations if both μ and σ are unknown. Solve the linear system algebraically and round your answers to a sensible degree of accuracy. This method, featured in pdfjoiner_(4)-216, is a must‑know for the Edexcel A‑Level exam.
在求解正态分布未知参数时，务必先标准化为 Z 值，谨慎处理尾部概率，如果 μ 和 σ 均未知则需建立两个方程。用代数方法求解线性方程组，并将答案四舍五入至合理精度。这一方法在 pdfjoiner_(4)-216 页中重点呈现，是 Edexcel A-Level 考试必须掌握的内容。
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