离散随机变量的期望值
如果您从离散随机变量中抽取一组观测值，您可以找到这些观测值的均值。随着观测值的数量增加，这个值将越来越接近离散随机变量的期望值。
If you take a set of observations from a discrete random variable, you can find the mean of those observations. As the number of observations increases, this value will get closer and closer to the expected value of the discrete random variable.
期望值的数学定义 / Mathematical Definition of Expected Value
离散随机变量X的期望值E(X)定义为：
\[E(X) = \sum x P(X = x)\]
其中，x是随机变量的可能取值，P(X = x)是对应的概率。
The expected value E(X) of a discrete random variable X is defined as:
\[E(X) = \sum x P(X = x)\]
Where x are the possible values of the random variable and P(X = x) are the corresponding probabilities.
例：公平骰子的期望值 / Example: Expected Value of a Fair Die
掷一枚公平的六面骰子，上表面出现的数字被建模为随机变量X。
A fair six-sided dice is rolled. The number that appears on the uppermost face is modelled by the random variable X.
a) 写出X的概率分布。
a) Write down the probability distribution of X.
b) 使用X的概率分布计算E(X)。
b) Use the probability distribution of X to calculate E(X).
a) 概率分布表格：
| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| P(X = x) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) |
由于骰子是公平的，每一面朝上的概率相等，都是\(\frac{1}{6}\)。
Since the dice is fair, each side is equally likely to end facing up, so the probability of any face ending up as the uppermost is \(\frac{1}{6}\).
b) 期望值计算：
b) The expected value is:
\[E(X) = \sum x P(X = x) = \frac{1}{6} + \frac{2}{6} + \frac{3}{6} + \frac{4}{6} + \frac{5}{6} + \frac{6}{6} = \frac{21}{6} = 3.5\]
有时概率分布中包含未知参数，需要使用期望值或其他条件来求解这些参数。
Sometimes probability distributions contain unknown parameters that need to be solved using expected values or other conditions.
例：求未知概率参数 / Example: Finding Unknown Probability Parameters
随机变量X具有以下概率分布：
The random variable X has a probability distribution as shown in the table.
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| P(X = x) | 0.1 | p | 0.3 | q | 0.2 |
给定E(X) = 3，求p和q的值。
Given that E(X) = 3, find the value of p and the value of q.
首先，所有概率之和必须等于1：
First, the sum of all probabilities must equal 1:
0.1 + p + 0.3 + q + 0.2 = 1
p + q + 0.6 = 1
p + q = 0.4 （方程1）
期望值计算：
Expected value calculation:
E(X) = 1×0.1 + 2×p + 3×0.3 + 4×q + 5×0.2 = 3
0.1 + 2p + 0.9 + 4q + 1 = 3
2p + 4q + 2 = 3
2p + 4q = 1 （方程2）
解方程组：
Solve the system of equations:
将方程1乘以2：2p + 2q = 0.8
方程2减去这个：(2p + 4q) - (2p + 2q) = 1 - 0.8
2q = 0.2
q = 0.1
代入方程1：p + 0.1 = 0.4
p = 0.3
如果X是离散随机变量，那么X²也是离散随机变量。您可以使用这个规则来确定X²的期望值。
If X is a discrete random variable, then X² is also a discrete random variable. You can use this rule to determine the expected value of X².
X²期望值的计算 / Expected Value of X²
\[E(X^2) = \sum x^2 P(X = x)\]
任何随机变量的函数也是随机变量。
Any function of a random variable is also a random variable.
例：计算X²的期望值 / Example: Calculating Expected Value of X²
离散随机变量X具有以下概率分布：
A discrete random variable X has the following probability distribution:
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| P(X = x) | \(\frac{12}{25}\) | \(\frac{6}{25}\) | \(\frac{4}{25}\) | \(\frac{3}{25}\) |
a) 写出X²的概率分布。
a) Write down the probability distribution for X².
b) 求E(X²)。
b) Find E(X²).
a) X²的概率分布：
a) The probability distribution for X² is:
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| x² | 1 | 4 | 9 | 16 |
| P(X = x) | \(\frac{12}{25}\) | \(\frac{6}{25}\) | \(\frac{4}{25}\) | \(\frac{3}{25}\) |
X可以取值1,2,3,4，所以X²可以取值1²,2²,3²,4²。因为X只取正值，所以P(X² = x²) = P(X = x)。
X can take values 1,2,3,4, so X² can take values 1²,2²,3²,4². Note that because X takes only positive values, P(X² = x²) = P(X = x).
b) E(X²) = Σx²P(X = x)
b) E(X²) = Σx²P(X = x)
= 1×(12/25) + 4×(6/25) + 9×(4/25) + 16×(3/25)
= (12/25) + (24/25) + (36/25) + (48/25)
= 120/25 = 4.8
注意 / Note：
通常E(X²) ≠ [E(X)]²。在这个例子中，E(X) = 1.92，(1.92)² = 3.6864 ≠ 4.8。
In general, E(X²) is not equal to [E(X)]². In this example, E(X) = 1.92 and (1.92)² = 3.6864 ≠ 4.8.
为了巩固对期望值的理解，以下是一些典型的练习题类型：
To consolidate understanding of expected values, here are some typical exercise types:
练习题类型示例 / Exercise Types Examples：
类型1：基本期望值计算
给定概率分布表，计算随机变量的期望值。
Type 1: Basic expected value calculation
Given a probability distribution table, calculate the expected value of the random variable.
类型2：求未知参数
给定期望值和部分概率，求未知概率参数。
Type 2: Finding unknown parameters
Given expected value and partial probabilities, find unknown probability parameters.
类型3：函数期望值
计算随机变量函数（如X²）的期望值。
Type 3: Expected value of functions
Calculate the expected value of functions of random variables (such as X²).
类型4：实际应用
在赌博、质量控制、金融等实际场景中应用期望值概念。
Type 4: Practical applications
Apply expected value concepts in gambling, quality control, finance, and other practical scenarios.