在实际应用中，我们经常需要根据给定的概率条件来求解正态分布的未知参数μ（均值）和σ（标准差）。这是统计学中参数估计的重要技能。
In practical applications, we often need to solve for unknown parameters μ (mean) and σ (standard deviation) of a normal distribution based on given probability conditions. This is an important skill in parameter estimation in statistics.
基本思路：使用标准化公式 Z = (X - μ)/σ 将问题转化为标准正态分布问题
Basic approach: Use the standardization formula Z = (X - μ)/σ to transform the problem into a standard normal distribution problem
当正态分布的方差已知，但均值未知时，我们可以通过给定的概率条件来求解均值。
When the variance of a normal distribution is known but the mean is unknown, we can solve for the mean through given probability conditions.
随机变量X ~ N(μ, 3²)
Random variable X ~ N(μ, 3²)
已知P(X > 20) = 0.20，求μ的值
Given P(X > 20) = 0.20, find the value of μ
解 / Solution:
1. 使用标准化公式：Z = (X - μ)/σ
1. Use standardization formula: Z = (X - μ)/σ
2. P(X > 20) = 0.20 → P(Z > (20 - μ)/3) = 0.20
3. 绘制Z的图表，p = 0.20
3. Draw a diagram for Z, p = 0.20
4. 由于P(Z > z) = 0.20 < 0.5，所以z > 0
4. Since P(Z > z) = 0.20 < 0.5, so z > 0
5. 在百分比点表中查找p = 0.20对应的z值
5. Look up the z-value corresponding to p = 0.20 in the percentage points table
6. z = 0.8416
7. 因此：(20 - μ)/3 = 0.8416
7. Therefore: (20 - μ)/3 = 0.8416
8. 解得：μ = 20 - 3 × 0.8416 = 17.4752
8. Solving: μ = 20 - 3 × 0.8416 = 17.4752
答案：μ = 17.5 (3 s.f.)
当正态分布的均值已知，但标准差未知时，我们可以通过给定的概率条件来求解标准差。
When the mean of a normal distribution is known but the standard deviation is unknown, we can solve for the standard deviation through given probability conditions.
机器制造的金属板宽度X ~ N(50, σ²)，单位为厘米
A machine makes metal sheets with width X ~ N(50, σ²) in centimeters
a) 已知P(X < 46) = 0.2119，求σ的值
a) Given P(X < 46) = 0.2119, find the value of σ
b) 求宽度的第90百分位数
b) Find the 90th percentile of the widths
解 / Solution:
a) 求σ的值
a) Find the value of σ
1. P(X < 46) = 0.2119
2. 使用标准化公式：P(Z < (46 - 50)/σ) = 0.2119
2. Use standardization formula: P(Z < (46 - 50)/σ) = 0.2119
3. P(Z < -4/σ) = 0.2119
4. 绘制Z的图表，p = 0.2119
4. Draw a diagram for Z, p = 0.2119
5. 由于P(Z < z) = 0.2119 < 0.5，所以z < 0
5. Since P(Z < z) = 0.2119 < 0.5, so z < 0
6. 使用对称性：P(Z < -0.80) = 0.2119
6. Use symmetry: P(Z < -0.80) = 0.2119
7. 因此：-4/σ = -0.80
7. Therefore: -4/σ = -0.80
8. 解得：σ = 4/0.80 = 5
8. Solving: σ = 4/0.80 = 5
答案：σ = 5
b) 求第90百分位数
b) Find the 90th percentile
1. 设第90百分位数为a，则P(X < a) = 0.9
1. Let the 90th percentile be a, then P(X < a) = 0.9
2. 现在X ~ N(50, 5²)
2. Now X ~ N(50, 5²)
3. P(Z < (a - 50)/5) = 0.9
4. 在主表中查找0.9对应的z值
4. Look up the z-value corresponding to 0.9 in the main table
5. z = 1.2816
6. 因此：(a - 50)/5 = 1.2816
6. Therefore: (a - 50)/5 = 1.2816
7. 解得：a = 50 + 5 × 1.2816 = 56.408
7. Solving: a = 50 + 5 × 1.2816 = 56.408
答案：a = 56.4 cm (3 s.f.)
当正态分布的均值和标准差都未知时，我们需要两个独立的概率条件来建立方程组求解。
When both the mean and standard deviation of a normal distribution are unknown, we need two independent probability conditions to establish a system of equations for solving.
随机变量X ~ N(μ, σ²)
Random variable X ~ N(μ, σ²)
已知P(X > 35) = 0.025 和 P(X < 15) = 0.1469
Given P(X > 35) = 0.025 and P(X < 15) = 0.1469
求μ和σ的值
Find the values of μ and σ
解 / Solution:
1. 建立两个方程：
1. Establish two equations:
P(X > 35) = 0.025 → P(Z > (35 - μ)/σ) = 0.025
P(X < 15) = 0.1469 → P(Z < (15 - μ)/σ) = 0.1469
2. 查找对应的z值：
2. Find corresponding z-values:
P(Z > z₁) = 0.025 → z₁ = 1.96 (从百分比点表)
P(Z > z₁) = 0.025 → z₁ = 1.96 (from percentage points table)
P(Z < z₂) = 0.1469 → z₂ = -1.05 (使用对称性)
P(Z < z₂) = 0.1469 → z₂ = -1.05 (using symmetry)
3. 建立方程组：
3. Establish system of equations:
(35 - μ)/σ = 1.96 → 1.96σ + μ = 35 ... (1)
(15 - μ)/σ = -1.05 → -1.05σ + μ = 15 ... (2)
4. 解方程组：
4. Solve the system:
(1) - (2): 3.01σ = 20
σ = 20/3.01 = 6.6445...
5. 代入(1)式：
5. Substitute into equation (1):
μ = 35 - 1.96 × 6.6445... = 21.976...
答案：σ = 6.64, μ = 22.0 (3 s.f.)
求解未知参数的标准步骤：
Standard steps for finding unknown parameters:
-
识别问题类型 / Identify problem type:
- 只求μ（σ已知）/ Only find μ (σ known)
- 只求σ（μ已知）/ Only find σ (μ known)
- 同时求μ和σ / Find both μ and σ
-
建立方程 / Establish equations:
- 使用标准化公式Z = (X - μ)/σ
- Use standardization formula Z = (X - μ)/σ
- 将概率条件转化为标准正态分布问题
- Transform probability conditions into standard normal distribution problems
-
查找z值 / Find z-values:
- 使用百分比点表或主表
- Use percentage points table or main table
- 注意z值的符号
- Pay attention to the sign of z-values
-
求解参数 / Solve for parameters:
- 单个参数：直接求解
- Single parameter: solve directly
- 两个参数：建立方程组求解
- Two parameters: establish system of equations and solve
-
验证答案 / Verify answer:
- 检查答案的合理性
- Check reasonableness of answers
- 代入原条件验证
- Substitute back into original conditions to verify
当给定的概率条件具有对称性时，可以利用对称性简化计算：
When given probability conditions have symmetry, we can use symmetry to simplify calculations:
• 如果P(X > a) = P(X < b) = p，则μ = (a + b)/2
• If P(X > a) = P(X < b) = p, then μ = (a + b)/2
• 利用对称性可以快速确定均值的位置
• Using symmetry can quickly determine the position of the mean
随机变量X ~ N(μ, σ²)
Random variable X ~ N(μ, σ²)
已知P(X > 15) = 0.20 和 P(X < 9) = 0.20
Given P(X > 15) = 0.20 and P(X < 9) = 0.20
求μ和σ的值
Find the values of μ and σ
解 / Solution:
1. 由于P(X > 15) = P(X < 9) = 0.20，具有对称性
1. Since P(X > 15) = P(X < 9) = 0.20, there is symmetry
2. 利用对称性：μ = (15 + 9)/2 = 12
2. Use symmetry: μ = (15 + 9)/2 = 12
3. 现在求σ：P(X > 15) = 0.20
3. Now find σ: P(X > 15) = 0.20
4. P(Z > (15 - 12)/σ) = 0.20
5. P(Z > 3/σ) = 0.20
6. 3/σ = 0.8416
7. σ = 3/0.8416 = 3.57
答案：μ = 12, σ = 3.57 (3 s.f.)
分位数问题是指给定某个百分位数，求对应的参数值。这类问题在实际应用中很常见。
Quantile problems refer to finding corresponding parameter values given a certain percentile. These problems are common in practical applications.
随机变量X ~ N(μ, σ²)
Random variable X ~ N(μ, σ²)
X的下四分位数是25，上四分位数是45
The lower quartile of X is 25 and the upper quartile is 45
求μ和σ的值
Find the values of μ and σ
解 / Solution:
1. 下四分位数：P(X < 25) = 0.25
1. Lower quartile: P(X < 25) = 0.25
2. 上四分位数：P(X < 45) = 0.75
2. Upper quartile: P(X < 45) = 0.75
3. 建立方程组：
3. Establish system of equations:
P(Z < (25 - μ)/σ) = 0.25 → (25 - μ)/σ = -0.6745
P(Z < (45 - μ)/σ) = 0.75 → (45 - μ)/σ = 0.6745
4. 解方程组：
4. Solve the system:
(25 - μ)/σ = -0.6745 → -0.6745σ + μ = 25 ... (1)
(45 - μ)/σ = 0.6745 → 0.6745σ + μ = 45 ... (2)
5. (2) - (1): 1.349σ = 20
6. σ = 20/1.349 = 14.83
7. μ = 45 - 0.6745 × 14.83 = 35.0
答案：μ = 35.0, σ = 14.8 (3 s.f.)
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方程建立错误 / Incorrect equation establishment: 忘记使用标准化公式
Forgetting to use the standardization formula -
z值符号错误 / Incorrect z-value signs: 没有正确判断z值的正负号
Not correctly determining the sign of z-values -
方程组求解错误 / Incorrect system solving: 在求解两个参数时出现计算错误
Calculation errors when solving for two parameters -
单位问题 / Unit issues: 忘记考虑单位的一致性
Forgetting to consider unit consistency -
近似值选择 / Approximation selection: 当概率值不在表中时，没有选择最接近的值
Not choosing the closest value when probability is not in the table
在求解未知参数时，一定要确保有足够的信息来唯一确定参数值。对于两个未知参数，通常需要两个独立的概率条件。
When solving for unknown parameters, make sure there is sufficient information to uniquely determine the parameter values. For two unknown parameters, two independent probability conditions are usually needed.
寻找未知参数μ和σ的技能在实际生活中有广泛应用：
The skill of finding unknown parameters μ and σ has wide applications in real life:
-
质量控制 / Quality Control: 根据产品合格率确定生产标准
Determining production standards based on product qualification rates -
医学研究 / Medical Research: 根据疾病分布确定正常值范围
Determining normal value ranges based on disease distribution -
教育评估 / Educational Assessment: 根据考试成绩分布确定评分标准
Determining grading standards based on exam score distribution -
金融风险管理 / Financial Risk Management: 根据历史数据确定风险参数
Determining risk parameters based on historical data