7.5 寻找未知μ和σ - 核心知识点总结
1. 基本概念 / Basic Concepts
2. 问题类型分类 / Problem Type Classification
| 问题类型 / Problem Type |
已知条件 / Given Conditions |
求解目标 / Target |
所需信息 / Required Information |
| 类型1 / Type 1 |
σ已知 / σ known |
求μ / Find μ |
1个概率条件 / 1 probability condition |
| 类型2 / Type 2 |
μ已知 / μ known |
求σ / Find σ |
1个概率条件 / 1 probability condition |
| 类型3 / Type 3 |
μ和σ都未知 / Both μ and σ unknown |
求μ和σ / Find μ and σ |
2个独立概率条件 / 2 independent probability conditions |
3. 求解未知均值μ / Finding Unknown Mean μ
标准步骤：
Standard steps:
- 使用标准化公式：P(X > a) = P(Z > (a - μ)/σ)
- Use standardization formula: P(X > a) = P(Z > (a - μ)/σ)
- 查找对应的z值
- Find corresponding z-value
- 建立方程：(a - μ)/σ = z
- Establish equation: (a - μ)/σ = z
- 求解μ：μ = a - zσ
- Solve for μ: μ = a - zσ
典型例子 / Typical Example
X ~ N(μ, 3²), P(X > 20) = 0.20
P(Z > (20 - μ)/3) = 0.20
z = 0.8416
(20 - μ)/3 = 0.8416
μ = 20 - 3 × 0.8416 = 17.5
4. 求解未知标准差σ / Finding Unknown Standard Deviation σ
标准步骤：
Standard steps:
- 使用标准化公式：P(X < a) = P(Z < (a - μ)/σ)
- Use standardization formula: P(X < a) = P(Z < (a - μ)/σ)
- 查找对应的z值（注意符号）
- Find corresponding z-value (pay attention to sign)
- 建立方程：(a - μ)/σ = z
- Establish equation: (a - μ)/σ = z
- 求解σ：σ = (a - μ)/z
- Solve for σ: σ = (a - μ)/z
典型例子 / Typical Example
X ~ N(50, σ²), P(X < 46) = 0.2119
P(Z < (46 - 50)/σ) = 0.2119
P(Z < -4/σ) = 0.2119
z = -0.80
-4/σ = -0.80
σ = 4/0.80 = 5
5. 同时求解μ和σ / Simultaneously Finding μ and σ
标准步骤：
Standard steps:
- 建立两个方程：
- Establish two equations:
- P(X > a₁) = p₁ → P(Z > (a₁ - μ)/σ) = p₁
- P(X < a₂) = p₂ → P(Z < (a₂ - μ)/σ) = p₂
- 查找对应的z值：z₁, z₂
- Find corresponding z-values: z₁, z₂
- 建立方程组：
- Establish system of equations:
- (a₁ - μ)/σ = z₁
- (a₂ - μ)/σ = z₂
- 解方程组求μ和σ
- Solve the system for μ and σ
典型例子 / Typical Example
X ~ N(μ, σ²)
P(X > 35) = 0.025 → z₁ = 1.96
P(X < 15) = 0.1469 → z₂ = -1.05
方程组 / System of equations:
(35 - μ)/σ = 1.96 → 1.96σ + μ = 35 ... (1)
(15 - μ)/σ = -1.05 → -1.05σ + μ = 15 ... (2)
(1) - (2): 3.01σ = 20 → σ = 6.64
μ = 35 - 1.96 × 6.64 = 22.0
6. 特殊情况处理 / Special Case Handling
对称性问题 / Symmetry Problems
当P(X > a) = P(X < b) = p时：
When P(X > a) = P(X < b) = p:
• 利用对称性：μ = (a + b)/2
• Use symmetry: μ = (a + b)/2
• 然后求σ：P(X > a) = p → σ = (a - μ)/z
• Then find σ: P(X > a) = p → σ = (a - μ)/z
分位数问题 / Quantile Problems
当给定分位数时：
When given quantiles:
• 下四分位数：P(X < Q₁) = 0.25
• Lower quartile: P(X < Q₁) = 0.25
• 上四分位数：P(X < Q₃) = 0.75
• Upper quartile: P(X < Q₃) = 0.75
• 建立方程组求解
• Establish system of equations to solve
7. 重要公式总结 / Important Formula Summary
8. 解题技巧 / Problem-Solving Tips
- 识别问题类型 - 确定是求μ、σ还是两者
- Identify problem type - Determine whether to find μ, σ, or both
- 绘制图表 - 帮助理解概率条件
- Draw diagrams - Help understand probability conditions
- 注意z值符号 - 正确判断z值的正负号
- Pay attention to z-value signs - Correctly determine the sign of z-values
- 检查方程数量 - 确保有足够的方程求解未知数
- Check number of equations - Ensure sufficient equations to solve unknowns
- 验证答案 - 代入原条件检查答案合理性
- Verify answers - Substitute back into original conditions to check reasonableness
9. 常见错误 / Common Mistakes
避免这些错误 / Avoid These Mistakes
- 忘记使用标准化公式
- Forgetting to use the standardization formula
- z值符号判断错误
- Incorrect z-value sign determination
- 方程组求解错误
- Incorrect system of equations solving
- 单位不一致
- Unit inconsistency
- 近似值选择不当
- Inappropriate approximation selection
- 没有验证答案的合理性
- Not verifying answer reasonableness
10. 实际应用 / Practical Applications
应用领域 / Application Areas
- 质量控制 / Quality Control: 根据合格率确定生产标准
- 医学研究 / Medical Research: 根据分布确定正常值范围
- 教育评估 / Educational Assessment: 根据成绩分布确定评分标准
- 金融风险管理 / Financial Risk Management: 根据历史数据确定风险参数
- 工程测量 / Engineering Measurement: 根据测量误差确定精度要求
11. 考试技巧 / Exam Tips
考试注意事项 / Exam Precautions
- 仔细阅读题目 - 确定已知条件和求解目标
- Read questions carefully - Determine given conditions and solution targets
- 分步骤解题 - 清晰地展示解题过程
- Solve step by step - Clearly show the solution process
- 检查计算 - 避免计算错误
- Check calculations - Avoid calculation errors
- 注意有效数字 - 按要求保留有效数字
- Pay attention to significant figures - Retain significant figures as required
- 验证答案 - 确保答案符合实际情况
- Verify answers - Ensure answers are realistic
12. 练习建议 / Practice Recommendations
提高技能的建议 / Suggestions for Skill Improvement
- 多做不同类型的题目 - 熟悉各种问题类型
- Practice different types of problems - Familiarize with various problem types
- 熟练掌握标准化公式 - 这是解题的基础
- Master the standardization formula - This is the foundation of problem solving
- 练习方程组求解 - 提高计算能力
- Practice system of equations solving - Improve calculation skills
- 注意特殊情况 - 掌握对称性等技巧
- Pay attention to special cases - Master techniques like symmetry
- 时间管理 - 在考试中合理分配时间
- Time management - Allocate time reasonably in exams