求解涉及随机变量的问题
假设你有两个随机变量：变量 \(X\) 和变量 \(Y = g(X)\)。如果 \(g\) 是一一对应的，并且你知道 \(Y\) 的期望值和方差，那么就有可能推导出 \(X\) 的期望值和方差。
Suppose you have two random variables: a variable \(X\), and a variable \(Y = g(X)\). If \(g\) is one-to-one, and you know the mean and variance of \(Y\), then it is possible to deduce the mean and variance of \(X\).
例 6.6.1 / Example 6.6.1：
\(X\) 是一个离散随机变量。离散随机变量 \(Y\) 定义为 \(Y = \frac{X - 150}{50}\)。
\(X\) is a discrete random variable. The discrete random variable \(Y\) is defined as \(Y = \frac{X - 150}{50}\).
已知 \(\mathrm{E}(Y) = 5.1\) 和 \(\operatorname{Var}(Y) = 2.5\)，求：
Given that \(\mathrm{E}(Y) = 5.1\) and \(\operatorname{Var}(Y) = 2.5\), find:
a) \(\mathrm{E}(X)\)
b) \(\operatorname{Var}(X)\)
解答 / Solution：
a) \(Y = \frac{X - 150}{50}\)，所以 \(X = 50Y + 150\)。重新排列得到 \(X\) 用 \(Y\) 表示的表达式。
a) \(Y = \frac{X - 150}{50}\), so \(X = 50Y + 150\). Rearrange to get an expression for \(X\) in terms of \(Y\).
\[\mathrm{E}(X) = \mathrm{E}(50Y + 150) = 50\mathrm{E}(Y) + 150 = 50 \times 5.1 + 150 = 255 + 150 = 405\]
使用 \(X\) 用 \(Y\) 表示的表达式。
Use your expression for \(X\) in terms of \(Y\).
b) 记住加150不影响方差，并且要用 \(50^2\) 乘以 \(\operatorname{Var}(Y)\) 得到 \(\operatorname{Var}(X)\)。
b) Remember that the '+150' does not affect the variance, and that you have to multiply \(\operatorname{Var}(Y)\) by \(50^2\) to get \(\operatorname{Var}(X)\).
\[\operatorname{Var}(X) = \operatorname{Var}(50Y + 150) = 50^2 \operatorname{Var}(Y) = 2500 \times 2.5 = 6250\]
随机变量函数反推的关键方法：
Key Methods for Inverse Problems with Random Variable Functions:
重要提醒 / Important Note：
证明 \(\mathrm{E}[(X - \mathrm{E}(X))^2] = \mathrm{E}(X^2) - [\mathrm{E}(X)]^2\) 和 \(\mathrm{E}(X + Y) = \mathrm{E}(X) + \mathrm{E}(Y)\)。
Show that \(\mathrm{E}[(X - \mathrm{E}(X))^2] = \mathrm{E}(X^2) - [\mathrm{E}(X)]^2\) and \(\mathrm{E}(X + Y) = \mathrm{E}(X) + \mathrm{E}(Y)\).