Solving Problems Involving Random Variables | 随机变量相关问题的求解方法

📚 Solving Problems Involving Random Variables | 随机变量相关问题的求解方法

Random variables form the foundation of further statistics. In this guide, we will explore how to define them, how to calculate probabilities, expectations and variances, and how to apply standard techniques to solve examination problems efficiently.

随机变量是进阶统计学的基石。本指南将探讨如何定义随机变量、如何计算概率、期望与方差，以及如何运用标准方法高效地解决考试题目。

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1. What Is a Random Variable? | 什么是随机变量？

A random variable is a variable whose possible values are numerical outcomes of a random experiment. It is usually denoted by a capital letter such as X or Y.

随机变量是一个变量，它的可能取值是随机试验的数值结果。通常用大写字母如 X 或 Y 表示。

There are two important questions to answer first: what values can the variable take, and what probabilities are assigned to those values?

需要先回答两个重要问题：这个变量可能取哪些值？这些值对应的概率是多少？

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2. Discrete and Continuous Random Variables | 离散型与连续型随机变量

A discrete random variable takes a finite or countably infinite set of values, such as 0, 1, 2, … . A continuous random variable takes any value in an interval, such as height or waiting time.

离散型随机变量取有限个或可数无穷多个值，例如 0, 1, 2, …。连续型随机变量可以在某个区间内取任意值，例如身高或等待时间。

The table below summarises the main differences.

下表总结了主要区别。

Property	Discrete	Continuous
Possible values	Countable list	Interval of real numbers
Probability representation	Probability mass function P(X = x)	Probability density function f(x)
Total probability	Σ P(X = x) = 1	∫ f(x) dx = 1 over the whole range
Point probability	Can be positive	Always zero at a single point

In an exam, always check whether the variable is discrete or continuous before choosing a method.

在考试中，务必先判断变量是离散型还是连续型，再选择方法。

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3. Probability Mass Function and Cumulative Distribution Function | 概率质量函数与累积分布函数

For a discrete random variable X, the probability mass function p(x) gives P(X = x). The cumulative distribution function F(x) is defined as P(X ≤ x).

对于离散型随机变量 X，概率质量函数 p(x) 给出 P(X = x)。累积分布函数 F(x) 定义为 P(X ≤ x)。

F(x) = P(X ≤ x) = Σ p(t), where the sum is over all t ≤ x.

To find probabilities such as P(X > 3), use the complement: P(X > 3) = 1 − F(3) for a discrete variable whose values are integers, because F(3) includes 0, 1, 2, 3.

要求 P(X > 3) 之类的概率时，可利用补事件：P(X > 3) = 1 − F(3)（对于取整数值的离散变量，因为 F(3) 包含 0、1、2、3）。

P(a < X ≤ b) = F(b) − F(a)

This identity is extremely useful when solving problems with cumulative probability tables.

这个恒等式在利用累积概率表解题时非常有用。

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4. Expectation and Variance of Discrete Random Variables | 离散型随机变量的期望与方差

The expectation E(X) is a weighted average of the possible values.

期望 E(X) 是可能取值的加权平均。

E(X) = Σ x p(x)

The variance measures the spread around the mean. The most efficient formula in problems is:

方差衡量取值在均值周围的分散程度。解题中最常用的公式是：

Var(X) = E(X²) − [E(X)]²

Always compute E(X²) = Σ x² p(x) first, then use the formula above.

一定要先计算 E(X²) = Σ x² p(x)，再代入上式。

For example, if X takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3, then E(X) = 1(0.2) + 2(0.5) + 3(0.3) = 2.1, and E(X²) = 1²(0.2) + 2²(0.5) + 3²(0.3) = 0.2 + 2 + 2.7 = 4.9, so Var(X) = 4.9 − 2.1² = 4.9 − 4.41 = 0.49.

例如，若 X 取 1、2、3 的概率分别为 0.2、0.5、0.3，则 E(X) = 1×0.2 + 2×0.5 + 3×0.3 = 2.1，E(X²) = 1²×0.2 + 2²×0.5 + 3²×0.3 = 0.2 + 2 + 2.7 = 4.9，所以 Var(X) = 4.9 − 2.1² =

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