6.6 Solving Problems Involving Random Variables

6.6.1 核心概念总结 / Core Concepts Summary

函数随机变量：

如果 \(X\) 是随机变量，\(g\) 是函数，那么 \(Y = g(X)\) 也是随机变量。可以利用 \(Y\) 的统计量推导出 \(X\) 的统计量。

Function Random Variables:

If \(X\) is a random variable and \(g\) is a function, then \(Y = g(X)\) is also a random variable. The statistics of \(Y\) can be used to deduce the statistics of \(X\).

线性变换反推：

对于线性函数 \(Y = aX + b\)，已知 \(Y\) 的期望值和方差可以直接推导出 \(X\) 的期望值和方差。

Linear Transformation Inversion:

For linear function \(Y = aX + b\), knowing the expected value and variance of \(Y\) allows direct deduction of the expected value and variance of \(X\).

反推的关键步骤：

Key Steps for Inversion:

函数反解：将 \(Y = g(X)\) 反解为 \(X = h(Y)\)。Function inversion: Solve \(Y = g(X)\) for \(X = h(Y)\).
期望值反推：\(\mathrm{E}(X) = \mathrm{E}(h(Y))\)。Expected value inversion: \(\mathrm{E}(X) = \mathrm{E}(h(Y))\).
方差反推：对于线性函数，\(\operatorname{Var}(X) = \frac{\operatorname{Var}(Y)}{a^2}\)。Variance inversion: For linear functions, \(\operatorname{Var}(X) = \frac{\operatorname{Var}(Y)}{a^2}\).
一一对应性：函数 \(g\) 必须是一一对应的。Bijectivity: Function \(g\) must be one-to-one.

6.6.2 反推公式总结 / Inversion Formulas Summary

线性变换 \(Y = aX + b\) 的反推公式：

Inversion formulas for linear transformation \(Y = aX + b\):

\[\mathrm{E}(X) = \frac{\mathrm{E}(Y) - b}{a}\]

\[\operatorname{Var}(X) = \frac{\operatorname{Var}(Y)}{a^2}\]

\[\sigma_X = \frac{\sigma_Y}{|a|}\]

标准化变换示例 / Standardization Example：

对于 \(Y = \frac{X - 150}{50}\)，如果 \(\mathrm{E}(Y) = 5.1\)，\(\operatorname{Var}(Y) = 2.5\)，则：

\[\mathrm{E}(X) = 50 \times 5.1 + 150 = 405\]

\[\operatorname{Var}(X) = 50^2 \times 2.5 = 6250\]

6.6.3 证明要点 / Proof Key Points

重要证明 / Important Proofs：

方差定义等价性：\(\mathrm{E}[(X - \mathrm{E}(X))^2] = \mathrm{E}(X^2) - [\mathrm{E}(X)]^2\)。
Variance definition equivalence: \(\mathrm{E}[(X - \mathrm{E}(X))^2] = \mathrm{E}(X^2) - [\mathrm{E}(X)]^2\).
期望值线性性：\(\mathrm{E}(X + Y) = \mathrm{E}(X) + \mathrm{E}(Y)\)。
Expected value linearity: \(\mathrm{E}(X + Y) = \mathrm{E}(X) + \mathrm{E}(Y)\).
方差缩放性：对于常数 \(a\)，\(\operatorname{Var}(aX) = a^2 \operatorname{Var}(X)\)。
Variance scaling: For constant \(a\), \(\operatorname{Var}(aX) = a^2 \operatorname{Var}(X)\).

证明技巧 / Proof Techniques：

利用期望值的线性性质进行反推。
Use the linearity of expected value for inversion.
注意常数项不影响方差。
Note that constant terms do not affect variance.
乘数因子影响方差的平方。
Multiplicative factors affect variance by their square.

6.6.4 应用要点总结 / Application Key Points

解题策略 / Problem-Solving Strategies：

识别线性关系：确认变换是否为线性形式。
Identify linear relationships: Confirm whether the transformation is linear.
求解反函数：将Y表示为X的函数，反解得到X表示为Y的函数。
Solve for inverse function: Express Y as a function of X, then solve for X as a function of Y.
应用线性性质：利用期望值和方差的线性性质进行计算。
Apply linearity properties: Use linearity properties of expected value and variance for calculations.
验证合理性：检查计算结果是否合理。
Verify reasonableness: Check if calculation results are reasonable.

常见错误提醒 / Common Mistakes Reminder：

忘记常数项不影响方差。
Forget that constant terms do not affect variance.
混淆期望值和方差的反推公式。
Confuse inversion formulas for expected value and variance.
忽略函数必须是一一对应的前提。
Ignore the prerequisite that the function must be one-to-one.

6.6.5 思维导图总结 / Mind Map Summary

随机变量函数反推知识体系：

核心概念	反推方法	公式应用	注意事项
• 函数随机变量 • 线性变换 • 一一对应性	• 函数反解 • 期望值反推 • 方差反推 • 标准差反推	• \(\mathrm{E}(X) = \frac{\mathrm{E}(Y)-b}{a}\) • \(\operatorname{Var}(X) = \frac{\operatorname{Var}(Y)}{a^2}\) • \(\sigma_X = \frac{\sigma_Y}{\|a\|}\)	• 函数必须一一对应 • 常数项不影响方差 • 乘数影响方差平方 • 验证结果合理性