第六章复习
本章节复习涵盖了第六章"离散随机变量"的所有核心概念和技能。通过这些综合练习，您将巩固对离散随机变量概率分布、期望值、方差、累积分布函数以及随机变量函数变换的理解。
This chapter review covers all the core concepts and skills from Chapter 6 "Discrete Random Variables". Through these comprehensive exercises, you will consolidate your understanding of discrete random variable probability distributions, expected values, variances, cumulative distribution functions, and transformations of random variables.
这些练习题按照技能水平组织，包括基础计算、概率求解、统计量计算以及实际应用问题。建议您在完成各小节学习后进行这些复习练习，以检验和提升您的理解。
These exercises are organized by skill level, including basic calculations, probability solving, statistical computations, and practical application problems. It is recommended that you complete these review exercises after finishing each section to test and improve your understanding.
取值有限或可数无限的随机变量，具有概率质量函数和累积分布函数。
Random variables that take finite or countably infinite values, with probability mass functions and cumulative distribution functions.
期望值是随机变量的"平均值"，方差测量其分布的散布程度。
Expected value is the "average value" of a random variable, variance measures the degree of dispersion of its distribution.
描述随机变量取各个可能值的概率，包括离散均匀分布等特殊形式。
Describes the probability that a random variable takes each possible value, including special forms like discrete uniform distributions.
随机变量的线性或非线性变换，其统计量的计算和反推问题。
Linear or nonlinear transformations of random variables, including calculation of their statistics and inverse problems.
\[\mathrm{E}(X) = \sum x \cdot P(X = x)\]
\[\operatorname{Var}(X) = \mathrm{E}(X^2) - [\mathrm{E}(X)]^2\]
\[F(x) = P(X \leq x)\]
\[\mathrm{E}(X) = \frac{n + 1}{2}, \quad \operatorname{Var}(X) = \frac{(n + 1)(n - 1)}{12}\]