求解涉及随机变量的问题
本节练习题旨在帮助您掌握随机变量函数反推的技巧。建议先尝试自行解答，然后点击"显示答案"按钮查看参考答案。每个练习题都有详细的解答步骤。
These exercises are designed to help you master the techniques for inverse problems with random variable functions. It is recommended to try solving them yourself first, then click the "Show Answer" button to view the reference answers. Each exercise includes detailed solution steps.
练习要点 / Exercise Key Points：
\(X\) 是一个离散随机变量。随机变量 \(Y\) 定义为 \(Y = 4X - 6\)。已知 \(\mathrm{E}(Y) = 2\) 和 \(\operatorname{Var}(Y) = 32\)，求：
a) \(\mathrm{E}(X)\)
b) \(\operatorname{Var}(X)\)
c) \(X\) 的标准差。
解答：
a) \(Y = 4X - 6\)，所以 \(X = \frac{Y + 6}{4}\)。
\(\mathrm{E}(X) = \frac{\mathrm{E}(Y) + 6}{4} = \frac{2 + 6}{4} = \frac{8}{4} = 2\)
b) \(\operatorname{Var}(X) = \frac{\operatorname{Var}(Y)}{4^2} = \frac{32}{16} = 2\)
c) 标准差 \(\sigma_X = \sqrt{\operatorname{Var}(X)} = \sqrt{2}\)
\(X\) 是一个离散随机变量。随机变量 \(Y\) 定义为 \(Y = \frac{4 - 3X}{2}\)。已知 \(\mathrm{E}(Y) = -1\) 和 \(\operatorname{Var}(Y) = 9\)，求：
a) \(\mathrm{E}(X)\)
b) \(\operatorname{Var}(X)\)
c) \(\mathrm{E}(X^2)\)。
解答：
a) \(Y = \frac{4 - 3X}{2}\)，所以 \(2Y = 4 - 3X\)，\(3X = 4 - 2Y\)，\(X = \frac{4 - 2Y}{3}\)。
\(\mathrm{E}(X) = \frac{4 - 2\mathrm{E}(Y)}{3} = \frac{4 - 2(-1)}{3} = \frac{4 + 2}{3} = \frac{6}{3} = 2\)
b) \(\operatorname{Var}(X) = \frac{\operatorname{Var}(Y)}{(-3)^2} = \frac{9}{9} = 1\)
c) \(\mathrm{E}(X^2) = \operatorname{Var}(X) + [\mathrm{E}(X)]^2 = 1 + 4 = 5\)
离散随机变量 \(X\) 的概率分布如下表所示：
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| P(X = x) | 0.3 | a | b | 0.2 |
随机变量 \(Y\) 定义为 \(Y = 2X + 3\)。已知 \(\mathrm{E}(Y) = 8\)，求a和b的值。
解答：
概率和为1：0.3 + a + b + 0.2 = 1
a + b = 0.5
\(\mathrm{E}(X) = 1(0.3) + 2a + 3b + 4(0.2) = 0.3 + 2a + 3b + 0.8 = 1.1 + 2a + 3b\)
\(\mathrm{E}(Y) = 2\mathrm{E}(X) + 3 = 8\)
2(1.1 + 2a + 3b) + 3 = 8
2.2 + 4a + 6b + 3 = 8
4a + 6b = 8 - 5.2 = 2.8
4a + 6b = 2.8
联立求解：
4a + 6b = 2.8
a + b = 0.5
从第二个方程：a = 0.5 - b
代入第一个：4(0.5 - b) + 6b = 2.8
2 - 4b + 6b = 2.8
2 + 2b = 2.8
2b = 0.8
b = 0.4
a = 0.5 - 0.4 = 0.1
离散随机变量 \(X\) 的概率分布如下表所示：
| x | 90° | 180° | 270° |
|---|---|---|---|
| P(X = x) | a | b | 0.3 |
随机变量 \(Y\) 定义为 \(Y = \sin X^\circ\)。
a) 求 \(\mathrm{E}(Y)\) 的可能取值范围。
b) 已知 \(\mathrm{E}(Y) = 0.2\)，写出a和b的值。
解答：
a) \(\sin 90^\circ = 1\), \(\sin 180^\circ = 0\), \(\sin 270^\circ = -1\)。
Y的可能值为-1, 0, 1。
\(\mathrm{E}(Y) = a(1) + b(0) + 0.3(-1) = a - 0.3\)
概率和：a + b + 0.3 = 1
b = 0.7 - a
所以 \(\mathrm{E}(Y) = a - 0.3\)
最小值：当a=0时，E(Y) = -0.3
最大值：当a=0.7时，E(Y) = 0.4
范围：[-0.3, 0.4]
b) \(\mathrm{E}(Y) = 0.2 = a - 0.3\)
a = 0.5
b = 0.7 - 0.5 = 0.2
通过练习您应该掌握：
熟练掌握这些内容将为您后续学习连续随机变量的函数变换以及统计推断打下坚实的基础。如果在练习过程中遇到困难，建议回顾教材内容中的概念解释和实例演示。
Mastering these concepts will lay a solid foundation for your subsequent study of function transformations for continuous random variables and statistical inference. If you encounter difficulties during practice, it is recommended to review the concept explanations and example demonstrations in the textbook content.