7.1 The Normal Distribution - 练习题

中文答案：

正态分布的主要特征：

1. 对称性：曲线关于均值μ对称

2. 单峰性：只有一个峰值，位于均值处

3. 渐近性：曲线向两端无限延伸，但永远不会与横轴相交

4. 面积性质：曲线下总面积为1

5. 拐点位置：在x = μ ± σ处有拐点

正态分布在统计学中重要的原因：

- 许多自然现象和社会现象近似服从正态分布

- 中心极限定理表明，大量独立随机变量的和近似服从正态分布

- 许多统计推断方法基于正态分布

- 计算便捷，有完善的理论基础

English Answer：

Main characteristics of normal distribution:

1. Symmetry: The curve is symmetric about the mean μ

2. Unimodality: Has a single peak at the mean

3. Asymptotic behavior: The curve extends infinitely in both directions but never intersects the horizontal axis

4. Area property: Total area under the curve is 1

5. Inflection points: Located at x = μ ± σ

Reasons for the importance of normal distribution in statistics:

- Many natural and social phenomena approximately follow normal distribution

- Central Limit Theorem states that the sum of many independent random variables approximately follows normal distribution

- Many statistical inference methods are based on normal distribution

- Convenient for calculation with a sound theoretical foundation

中文答案：

正态分布的概率密度函数为：\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]

参数含义：

- μ（mu）：均值，决定分布的中心位置

- σ（sigma）：标准差，决定分布的形状（分散程度）

- π：圆周率，约等于3.14159

- e：自然对数的底数，约等于2.71828

English Answer：

The probability density function of normal distribution is: \[ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]

Meaning of parameters:

- μ (mu): Mean, determines the central position of the distribution

- σ (sigma): Standard deviation, determines the shape (spread) of the distribution

- π: Pi, approximately 3.14159

- e: Base of natural logarithm, approximately 2.71828

中文答案：

已知μ = 20°C，σ = 5°C

a) 15°C到25°C是μ ± σ范围，根据经验法则，约68%的气温在此范围内。

b) 10°C到30°C是μ ± 2σ范围，根据经验法则，约95%的气温在此范围内。

c) 5°C到35°C是μ ± 3σ范围，根据经验法则，约99.7%的气温在此范围内。

English Answer：

Given μ = 20°C, σ = 5°C

a) 15°C to 25°C is the range μ ± σ. According to the empirical rule, approximately 68% of temperatures are in this range.

b) 10°C to 30°C is the range μ ± 2σ. According to the empirical rule, approximately 95% of temperatures are in this range.

c) 5°C to 35°C is the range μ ± 3σ. According to the empirical rule, approximately 99.7% of temperatures are in this range.

中文答案：

已知μ = 75分，σ = 10分

a) 65分是μ - σ。根据经验法则，约68%的数据在μ ± σ范围内，因此约32%的数据在μ ± σ范围外，其中约16%在μ - σ以下，约16%在μ + σ以上。所以得分在65分以上的学生约占84%（50% + 34%）。

b) 85分是μ + σ。得分在85分以上的学生约占16%。

c) 95分是μ + 2σ。根据经验法则，约95%的数据在μ ± 2σ范围内，因此约5%的数据在μ ± 2σ范围外，其中约2.5%在μ + 2σ以上。所以得分在95分以上的学生约占2.5%。

English Answer：

Given μ = 75, σ = 10

a) 65 is μ - σ. According to the empirical rule, about 68% of data are within μ ± σ, so about 32% are outside this range, with approximately 16% below μ - σ and 16% above μ + σ. Therefore, about 84% (50% + 34%) of students scored above 65.

b) 85 is μ + σ. Approximately 16% of students scored above 85.

c) 95 is μ + 2σ. According to the empirical rule, about 95% of data are within μ ± 2σ, so about 5% are outside this range, with approximately 2.5% above μ + 2σ. Therefore, about 2.5% of students scored above 95.

中文答案：

已知μ = 10厘米，σ = 0.2厘米

a) 合格范围是9.6厘米到10.4厘米，即μ ± 2σ。根据经验法则，约95%的数据在此范围内，因此合格率约为95%。

b) 不合格率约为5%，因此1000个零件中约有1000 × 5% = 50个不合格。

English Answer：

Given μ = 10 cm, σ = 0.2 cm

a) The qualified range is 9.6 cm to 10.4 cm, which is μ ± 2σ. According to the empirical rule, approximately 95% of data are within this range, so the pass rate is about 95%.

b) The unqualified rate is about 5%, so among 1000 parts, approximately 1000 × 5% = 50 will be unqualified.

中文答案：

已知μ = 8000元，σ = 1200元

a) 6800元到9200元是μ ± σ范围，根据经验法则，约68%的员工月薪在此范围内。

b) 5600元到10400元是μ ± 2σ范围，根据经验法则，约95%的员工月薪在此范围内。

c) 10400元是μ + 2σ。根据经验法则，约2.5%的员工月薪超过10400元。因此500名员工中约有500 × 2.5% = 12.5 ≈ 13名员工月薪超过10400元。

English Answer：

Given μ = 8000 yuan, σ = 1200 yuan

a) 6800 yuan to 9200 yuan is the range μ ± σ. According to the empirical rule, approximately 68% of employees have monthly salaries in this range.

b) 5600 yuan to 10400 yuan is the range μ ± 2σ. According to the empirical rule, approximately 95% of employees have monthly salaries in this range.

c) 10400 yuan is μ + 2σ. According to the empirical rule, approximately 2.5% of employees have monthly salaries exceeding 10400 yuan. Therefore, among 500 employees, approximately 500 × 2.5% = 12.5 ≈ 13 have monthly salaries exceeding 10400 yuan.

中文答案：

a) X的概率密度函数为：\[ f(x) = \frac{1}{10\sqrt{2\pi}} e^{-\frac{(x-50)^2}{200}} \]

b) 均值μ = 50，标准差σ = 10（因为方差σ² = 100）

c) P(40 ≤ X ≤ 60) 是μ ± σ范围的概率，约为0.68

d) P(X ≥ 70) 是X大于μ + 2σ的概率，约为0.025

English Answer：

a) The probability density function of X is: \[ f(x) = \frac{1}{10\sqrt{2\pi}} e^{-\frac{(x-50)^2}{200}} \]

b) Mean μ = 50, standard deviation σ = 10 (since variance σ² = 100)

c) P(40 ≤ X ≤ 60) is the probability within μ ± σ range, approximately 0.68

d) P(X ≥ 70) is the probability of X being greater than μ + 2σ, approximately 0.025

中文答案：

a) N(15, 9)的中心位置更靠右，因为其均值μ = 15大于N(10, 4)的均值μ = 10。

b) N(15, 9)的形状更分散，因为其方差σ² = 9，标准差σ = 3，大于N(10, 4)的方差σ² = 4，标准差σ = 2。

c) 第一个分布中距离均值2个标准差的区间：μ ± 2σ = 10 ± 2×2 = (6, 14)

d) 第二个分布中距离均值3个标准差的区间：μ ± 3σ = 15 ± 3×3 = (6, 24)

English Answer：

a) N(15, 9) has its center positioned more to the right because its mean μ = 15 is greater than the mean μ = 10 of N(10, 4).

b) N(15, 9) has a more spread-out shape because its variance σ² = 9 and standard deviation σ = 3 are greater than those of N(10, 4) (variance σ² = 4, standard deviation σ = 2).

c) The interval 2 standard deviations away from the mean in the first distribution: μ ± 2σ = 10 ± 2×2 = (6, 14)

d) The interval 3 standard deviations away from the mean in the second distribution: μ ± 3σ = 15 ± 3×3 = (6, 24)