离散随机变量
随机变量是一个变量，其值依赖于随机事件的结局。随机变量用大写字母表示，例如X或Y，而随机变量可能取的特定值用相应的小写字母表示，例如x或y。随机变量X取特定值x的概率记为P(X = x)。
A random variable is a variable whose value depends on the outcome of a random event. Random variables are written using upper case letters, for example X or Y. The particular values that the random variable can take are written using equivalent lower case letters, for example x or y. The probability that the random variable X takes a particular value x is written as P(X = x).
随机变量用大写字母表示，其可能取的特定值用相应的小写字母表示。随机变量X取特定值x的概率记为P(X = x)。
Random variables are denoted by uppercase letters, while their possible values are denoted by the corresponding lowercase letters. The probability that random variable X takes a particular value x is denoted as P(X = x).
随机变量表示法 / Random Variable Notation
随机变量：\(X, Y, Z\)
随机变量取值：\(x, y, z\)
概率表示：\(P(X = x)\)
Random variables: \(X, Y, Z\)
Values: \(x, y, z\)
Probability: \(P(X = x)\)
重要表示法 / Important Notation：
随机变量用大写字母表示（如X、Y、Z），而其可能的取值用相应的小写字母表示（如x、y、z）。这种约定有助于区分随机变量本身与其取值。
Random variables are denoted by uppercase letters (such as X, Y, Z), while their possible values are denoted by the corresponding lowercase letters (such as x, y, z). This convention helps distinguish between the random variable itself and its values.
随机变量可能取值的集合称为样本空间。随机变量的取值范围决定了其样本空间。
The range of values that a random variable can take is called its sample space. The sample space is determined by the possible values of the random variable.
变量可以取一系列特定的值。如果变量只能取某些数值，则称其为离散的。如果变量的结果在实验进行之前是未知的，则称其为随机的。
A variable can take any of a range of specific values. The variable is discrete if it can take only certain numerical values. The variable is random if the outcome is not known until the experiment is carried out.
变量特征总结 / Summary of Variable Characteristics：
概率分布完全描述了样本空间中任何结果的概率。对于离散随机变量，概率分布可以用多种不同的方式描述。
A probability distribution fully describes the probability of any outcome in the sample space. The probability distribution for a discrete random variable can be described in a number of different ways.
以随机变量X = "掷一枚公平骰子得到的点数"为例，可以用以下方式描述其概率分布：
For example, take the random variable X = "score when a fair dice is rolled". It can be described:
骰子概率分布示例 / Dice Probability Distribution Example：
概率函数表示 / Probability Function Representation：
\(P(X = x) = \frac{1}{6}\), 对于 \(x = 1, 2, 3, 4, 5, 6\)
\(P(X = x) = \frac{1}{6}\), for \(x = 1, 2, 3, 4, 5, 6\)
表格表示 / Table Representation：
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| \(P(X = x)\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) |
图表表示 / Diagram Representation：
当所有概率相等时，这种分布称为离散均匀分布。
When all of the probabilities are the same, as in this example, the distribution is known as a discrete uniform distribution.
判断一个变量是否为离散随机变量需要考虑其取值特征和随机性特征。
Determining whether a variable is a discrete random variable requires considering both its value characteristics and randomness characteristics.
例：判断随机变量类型 / Example: Determining Random Variable Types
判断以下变量是否为离散随机变量：
Write down whether or not each of the following is a discrete random variable:
a) 一组学生中身高的平均值
a) The average height of a group of students
b) 掷一枚硬币直到出现正面所需的次数
b) The number of times a coin is tossed before a head appears
c) 一年的月数
c) The number of months in a year
解答 / Solution：
a) 不是离散随机变量，因为身高是在连续尺度上测量的。
a) Is not a discrete random variable as height is measured on a continuous scale.
b) 是离散随机变量，因为它是实验结果的计数。
b) Is a discrete random variable as it is a number that is the result of an experiment.
c) 不是离散随机变量，因为它不随实验变化且不是实验的结果。
c) Is not a discrete random variable as it does not vary and is not the result of an experiment.
掷骰子是离散随机变量的经典例子。让我们通过掷三枚硬币的实验来理解概率分布的建立。
Dice rolling is a classic example of discrete random variables. Let's understand the establishment of probability distributions through an experiment of tossing three coins.
例：三枚硬币实验 / Example: Three Coins Experiment
掷三枚公平的硬币。随机变量X定义为出现的正面数。
Three fair coins are tossed. A random variable X is defined as the number of heads that appear when the three coins are tossed.
实验所有可能结果 / All Possible Outcomes：
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
X的样本空间为 {0, 1, 2, 3}
概率分布表格 / Probability Distribution Table：
| 正面数 x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X = x) | \(\frac{1}{8}\) | \(\frac{3}{8}\) | \(\frac{3}{8}\) | \(\frac{1}{8}\) |
概率函数表示 / Probability Function：
X的取值：x = 1, 2
X的样本空间：{0, 1, 2, 3}
重要性质 / Important Property：
随机变量X的所有可能取值的概率之和等于1：
\(\sum P(X = x) = 1\) 对于所有x
The sum of the probabilities of all outcomes of an event adds up to 1. For a random variable X, you can write \(\sum P(X = x) = 1\) for all x.
偏斜骰子实验展示了如何处理非均匀概率分布的情况。这是一个更复杂的离散随机变量例子。
The biased dice experiment demonstrates how to handle non-uniform probability distributions. This is a more complex example of discrete random variables.
例：偏斜四面骰子 / Example: Biased Four-Sided Dice
一个偏斜的四面骰子，标有数字1、2、3、4。骰子底面上的数字被建模为随机变量X。
A biased four-sided dice with faces numbered 1, 2, 3 and 4 is rolled. The number on the bottommost face is modelled as a random variable X.
问题 / Problem：
给定 P(X = x) = k/x，求k的值。
Given that P(X = x) = k/x, find the value of k.
概率分布将为：
The probability distribution will be:
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| P(X = x) | k/1 | k/2 | k/3 | k/4 |
由于这是一个概率分布：
Since this is a probability distribution,
\(\sum P(X = x) = 1\)
\(k(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}) = 1\)
\(k(\frac{12 + 6 + 4 + 3}{12}) = 1\)
\(k = \frac{12}{25}\)
旋转器实验展示了更复杂的概率计算，包括条件概率和序列实验。
The spinner experiment demonstrates more complex probability calculations, including conditional probability and sequential experiments.
例：公平旋转器实验 / Example: Fair Spinner Experiment
一个公平的旋转器被旋转，直到它停在红色上，或已经被旋转了四次。随机变量S表示旋转的次数。
A fair spinner is spun until it lands on red or has been spun four times in total. Find the probability distribution of the random variable S, the number of times the spinner is spun.
P(S = 1) 是旋转器第一次就停在红色的概率：\(\frac{2}{5}\)
P(S = 1) is the probability that the spinner lands on red the first time: \(\frac{2}{5}\)
如果旋转器第二次停在红色，它第一次必须停在蓝色：\(\frac{3}{5} \times \frac{2}{5} = \frac{6}{25}\)
If the spinner lands on red on the second spin it must have landed on blue on the first spin: \(\frac{3}{5} \times \frac{2}{5} = \frac{6}{25}\)
类似地，第三次停在红色：\(\frac{3}{5} \times \frac{3}{5} \times \frac{2}{5} = \frac{18}{125}\)
Likewise for landing on red on the third spin: \(\frac{3}{5} \times \frac{3}{5} \times \frac{2}{5} = \frac{18}{125}\)
实验在4次旋转后停止：\(P(S = 4) = 1 - (\frac{2}{5} + \frac{6}{25} + \frac{18}{125}) = \frac{27}{125}\)
The experiment stops after 4 spins: \(P(S = 4) = 1 - (\frac{2}{5} + \frac{6}{25} + \frac{18}{125}) = \frac{27}{125}\)
概率分布表格 / Probability Distribution Table：
| s | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| P(S = s) | \(\frac{2}{5}\) | \(\frac{6}{25}\) | \(\frac{18}{125}\) | \(\frac{27}{125}\) |
为了巩固对离散随机变量的理解，以下是一些典型的练习题类型：
To consolidate understanding of discrete random variables, here are some typical exercise types:
练习题类型示例 / Exercise Types Examples：
类型1：判断随机变量类型
判断一组学生身高的平均值是否为离散随机变量。
Type 1: Determining variable types
Determine whether the average height of a group of students is a discrete random variable.
类型2：建立概率分布表
掷四次骰子，记录6出现的次数Y，建立概率分布表。
Type 2: Creating probability distribution tables
Roll a die four times and record Y as the number of times 6 appears. Create a probability distribution table.
类型3：计算概率
给定概率分布表，计算P(X > 15)或P(3 ≤ X < 11)等。
Type 3: Calculating probabilities
Given a probability distribution table, calculate P(X > 15) or P(3 ≤ X < 11), etc.
类型4：求未知参数
给定概率函数形式，求常数k的值。
Type 4: Finding unknown parameters
Given a probability function form, find the value of constant k.
注意事项 / Important Notes：
实用技巧 / Practical Tips：