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6.5 Expected Value and Variance of a Function of X

随机变量函数的期望值和方差 - 彩色打印版

一、核心概念 / Core Concepts

1. 函数期望值 / Expected Value of Functions

如果X是离散随机变量，g是函数，那么g(X)也是离散随机变量。
If X is a discrete random variable, and g is a function, then g(X) is also a discrete random variable.
数学定义：E(g(X)) = Σg(x)P(X = x)
Mathematical definition: E(g(X)) = Σg(x)P(X = x)
这是E(X²)公式的更一般形式。
This is a more general version of the formula for E(X²).

2. 线性变换规则 / Linear Transformation Rules

对于简单函数，如加法和乘以常数，可以使用特定规则简化计算。
For simple functions, such as addition and multiplication by a constant, specific rules can be used to simplify calculations.
  • 期望值规则：E(aX + b) = aE(X) + b（其中a和b是常数）
  • Expected Value Rules: E(aX + b) = aE(X) + b (where a and b are constants)
  • 随机变量和的期望值：E(X + Y) = E(X) + E(Y)
  • Expected Value of Sum: E(X + Y) = E(X) + E(Y)

3. 方差变换规则 / Variance Transformation Rules

可以使用类似的规则来简化某些随机变量函数的方差计算。
Similar rules can be used to simplify variance calculations for some functions of random variables.
  • 方差规则：Var(aX + b) = a²Var(X)（其中a和b是常数）
  • Variance Rules: Var(aX + b) = a²Var(X) (where a and b are constants)
  • 重要提示：常数项不影响方差，只有系数a会影响方差的大小
  • Important Note: Constant terms do not affect variance, only the coefficient a affects the magnitude of variance.

二、重要公式 / Important Formulas

1. 函数期望值公式 / Function Expected Value Formula

\[E(g(X)) = \sum g(x) P(X = x)\]
随机变量X的函数g(X)的期望值。
The expected value of function g(X) of random variable X.

2. 线性变换期望值 / Linear Transformation Expected Value

\[E(aX + b) = aE(X) + b\]
其中a和b是常数。
Where a and b are constants.

3. 线性变换方差 / Linear Transformation Variance

\[Var(aX + b) = a^2 Var(X)\]
其中a和b是常数，常数项不影响方差。
Where a and b are constants, constant terms do not affect variance.

4. 随机变量和的期望值 / Expected Value of Sum of Random Variables

\[E(X + Y) = E(X) + E(Y)\]
如果X和Y是随机变量。
If X and Y are random variables.

三、经典例子 / Classic Examples

例11：线性变换 / Example 11: Linear Transformation

离散随机变量X具有概率分布：
A discrete random variable X has the 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Y的概率分布，其中Y = 2X + 1
a) Write down the probability distribution for Y, where Y = 2X + 1
b) 求E(Y)
b) Find E(Y)
c) 计算E(X)并验证E(Y) = 2E(X) + 1
c) Compute E(X) and verify that E(Y) = 2E(X) + 1
解答：
Solution:
a) Y的概率分布：
a) The probability distribution for Y is:
x 1 2 3 4
y = 2x + 1 3 5 7 9
P(Y = y) \(\frac{12}{25}\) \(\frac{6}{25}\) \(\frac{4}{25}\) \(\frac{3}{25}\)
b) E(Y) = 3×(12/25) + 5×(6/25) + 7×(4/25) + 9×(3/25) = 121/25 = 4.84
b) E(Y) = 3×(12/25) + 5×(6/25) + 7×(4/25) + 9×(3/25) = 121/25 = 4.84
c) E(X) = 1×(12/25) + 2×(6/25) + 3×(4/25) + 4×(3/25) = 48/25 = 1.92
c) E(X) = 1×(12/25) + 2×(6/25) + 3×(4/25) + 4×(3/25) = 48/25 = 1.92
验证：2E(X) + 1 = 2×1.92 + 1 = 4.84 = E(Y) ✓
Verification: 2E(X) + 1 = 2×1.92 + 1 = 4.84 = E(Y) ✓

例12：使用已知统计量 / Example 12: Using Known Statistics

随机变量X有E(X) = 4和Var(X) = 3。求：
A random variable X has E(X) = 4 and Var(X) = 3. Find:
a) E(3X) b) E(X - 2) c) Var(3X) d) Var(X - 2) e) E(X²)
a) E(3X) b) E(X - 2) c) Var(3X) d) Var(X - 2) e) E(X²)
解答：
Solution:
a) E(3X) = 3E(X) = 3×4 = 12
a) E(3X) = 3E(X) = 3×4 = 12
b) E(X - 2) = E(X) - 2 = 4 - 2 = 2
b) E(X - 2) = E(X) - 2 = 4 - 2 = 2
c) Var(3X) = 3²Var(X) = 9×3 = 27
c) Var(3X) = 3²Var(X) = 9×3 = 27
d) Var(X - 2) = Var(X) = 3
d) Var(X - 2) = Var(X) = 3
e) E(X²) = Var(X) + [E(X)]² = 3 + 4² = 19
e) E(X²) = Var(X) + [E(X)]² = 3 + 4² = 19

四、练习题 / Practice Exercises

问题 1 / Question 1

随机变量X具有以下概率分布：
The random variable X has a probability distribution given by:
x 1 2 3 4
P(X = x) 0.1 0.3 0.2 0.4
a) 写出Y的概率分布，其中Y = 2X - 3
a) Write down the probability distribution for Y where Y = 2X - 3
b) 求E(Y)
b) Find E(Y)
c) 计算E(X)并验证E(2X - 3) = 2E(X) - 3
c) Calculate E(X) and verify that E(2X - 3) = 2E(X) - 3
答案空间 / Answer Space:
Answer Space:

问题 2 / Question 2

随机变量X有E(X) = 1和Var(X) = 2。求：
The random variable X has E(X) = 1 and Var(X) = 2. Find:
a) E(8X) b) E(X + 3) c) Var(X + 3) d) Var(3X) e) Var(1 - 2X) f) E(X²)
a) E(8X) b) E(X + 3) c) Var(X + 3) d) Var(3X) e) Var(1 - 2X) f) E(X²)
答案空间 / Answer Space:
Answer Space:

问题 3 / Question 3

在一个太空主题的棋盘游戏中，玩家每次绕棋盘一圈时掷一枚公平的六面骰子。玩家收集200分，加上骰子分数的100倍。
In a space-themed board game, players roll a fair, six-sided dice each time they make it around the board. Players collect 200 points, plus 100 times the score on the dice.
a) 写出E(X)，其中X是骰子分数
a) Write down E(X), where X is the dice score
b) 用X表示Y，其中Y是玩家获得的分数
b) Write Y in terms of X, where Y is the points received by the player
c) 求玩家每次绕棋盘一圈时获得的期望分数
c) Find the expected number of points a player receives each time they make it around the board
答案空间 / Answer Space:
Answer Space:

五、重要提示 / Important Tips

学习与应用要点 / Key Points for Learning and Application

  • 线性变换性质：线性变换保持期望值的线性性质，但方差会按系数平方缩放。
  • Linear Transformation Properties: Linear transformations preserve the linearity of expected values, but variance scales by the square of the coefficient.
  • 常数项影响：常数项只影响期望值，不影响方差。
  • Constant Term Effects: Constant terms only affect expected values, not variance.
  • 复合函数处理：对于复合函数，需要构造新的概率分布，然后应用期望值公式。
  • Composite Function Handling: For composite functions, construct new probability distributions and then apply expected value formulas.
  • 验证技巧：使用线性变换规则验证直接计算的结果。
  • Verification Techniques: Use linear transformation rules to verify results from direct calculations.

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