随机变量函数的期望值和方差
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。您可以使用以下公式计算g(X)的期望值：
If X is a discrete random variable, and g is a function, then g(X) is also a discrete random variable. You can calculate the expected value of g(X) using the formula:
函数期望值公式 / Function Expected Value Formula
\[E(g(X)) = \sum g(x) P(X = x)\]
这是E(X²)公式的更一般形式。
This is a more general version of the formula for E(X²).
对于简单函数，如加法和乘以常数，您可以学习以下规则：
For simple functions, such as addition and multiplication by a constant, you can learn the following rules:
线性变换规则 / Linear Transformation Rules
如果X是随机变量，a和b是常数，则：
\[E(aX + b) = aE(X) + b\]
If X is a random variable and a and b are constants, then:
\[E(aX + b) = aE(X) + b\]
随机变量和的期望值 / Expected Value of Sum of Random Variables
如果X和Y是随机变量，则：
\[E(X + Y) = E(X) + E(Y)\]
If X and Y are random variables, then:
\[E(X + Y) = E(X) + E(Y)\]
您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：
You can use a similar rule to simplify variance calculations for some functions of random variables:
线性变换方差公式 / Linear Transformation Variance Formula
如果X是随机变量，a和b是常数，则：
\[Var(aX + b) = a^2 Var(X)\]
If X is a random variable and a and b are constants, then:
\[Var(aX + b) = a^2 Var(X)\]
重要提示 / Important Note
注意常数项不影响方差，只有系数a会影响方差的大小。
Note that constant terms do not affect variance, only the coefficient a affects the magnitude of variance.
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) | \(\frac{12}{25}\) | \(\frac{6}{25}\) | \(\frac{4}{25}\) | \(\frac{3}{25}\) |
当x = 1时，y = 2×1 + 1 = 3；当x = 2时，y = 2×2 + 1 = 5，等等。
When x = 1, y = 2×1 + 1 = 3; when x = 2, y = 2×2 + 1 = 5, etc.
注意与X相关的概率仍然被使用，例如P(X = 3) = P(Y = 7)。
Notice how the probabilities relating to X are still being used, for example, P(X = 3) = P(Y = 7).
b) E(Y) = ΣyP(Y = y)
b) E(Y) = ΣyP(Y = y)
= 3×(12/25) + 5×(6/25) + 7×(4/25) + 9×(3/25)
= (36/25) + (30/25) + (28/25) + (27/25)
= 121/25 = 4.84
c) E(X) = ΣxP(X = x) = 1×(12/25) + 2×(6/25) + 3×(4/25) + 4×(3/25)
c) E(X) = ΣxP(X = x) = 1×(12/25) + 2×(6/25) + 3×(4/25) + 4×(3/25)
= (12/25) + (12/25) + (12/25) + (12/25) = 48/25 = 1.92
因此，2E(X) + 1 = 2×1.92 + 1 = 4.84
Therefore, 2E(X) + 1 = 2×1.92 + 1 = 4.84
这确认了E(Y) = 2E(X) + 1
This confirms that E(Y) = 2E(X) + 1
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)
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
重新排列Var(X) = E(X²) - [E(X)]²得到E(X²) = Var(X) + [E(X)]²
Rearrange Var(X) = E(X²) - [E(X)]² to get E(X²) = Var(X) + [E(X)]²
例13：硬币投掷问题 / Example 13: Coin Tossing Problem
投掷两枚公平的10分硬币。随机变量X（分）表示正面朝上的硬币总价值。
Two fair 10-cent coins are tossed. The random variable X cents represents the total value of the coins that land heads up.
a) 求E(X)和Var(X)。
a) Find E(X) and Var(X).
随机变量S和T定义如下：
The random variables S and T are defined as follows:
\[S = X - 10 \text{ and } T = \frac{1}{2}X - 5\]
b) 证明E(S) = E(T)。
b) Show that E(S) = E(T).
c) 求Var(S)和Var(T)。
c) Find Var(S) and Var(T).
d) 对大量S和T观测值的可能差异或相似性进行评论。
d) Comment on any likely differences or similarities.
解答 / Solution：
a) X的概率分布：
a) The probability distribution of X is:
| x | 0 | 10 | 20 |
|---|---|---|---|
| P(X = x) | \(\frac{1}{4}\) | \(\frac{1}{2}\) | \(\frac{1}{4}\) |
E(X) = 10（通过观察）
E(X) = 10 (by inspection)
Var(X) = E(X²) - [E(X)]²
Var(X) = E(X²) - [E(X)]²
= 0²×(1/4) + 10²×(1/2) + 20²×(1/4) - 10²
= 0 + 50 + 100 - 100 = 50
b) E(S) = E(X - 10) = E(X) - 10 = 10 - 10 = 0
b) E(S) = E(X - 10) = E(X) - 10 = 10 - 10 = 0
E(T) = E(½X - 5) = ½E(X) - 5 = ½×10 - 5 = 0
E(T) = E(½X - 5) = ½E(X) - 5 = ½×10 - 5 = 0
c) Var(S) = Var(X) = 50
c) Var(S) = Var(X) = 50
Var(T) = (½)²Var(X) = 50/4 = 12.5
Var(T) = (½)²Var(X) = 50/4 = 12.5
d) 两组观测值的均值都应该接近零。S的观测值将比T的观测值更加分散。
d) The means of both sets of observations should be close to zero. The observed values of S will be more spread out than the observed values of T.
例14：三角函数 / Example 14: Trigonometric Functions
随机变量X具有以下概率分布：
The random variable X has the following probability distribution:
| x | 0° | 30° | 60° | 90° |
|---|---|---|---|---|
| P(X = x) | 0.4 | 0.2 | 0.1 | 0.3 |
计算E(sin X)。
Calculate E(sin X).
解答 / Solution：
sin X的分布：
The distribution of sin X is:
| sin x | 0 | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | 1 |
|---|---|---|---|---|
| P(X = x) | 0.4 | 0.2 | 0.1 | 0.3 |
E(sin X) = Σsin x P(X = x)
E(sin X) = Σsin x P(X = x)
= 0×0.4 + (1/2)×0.2 + (√3/2)×0.1 + 1×0.3
= 0 + 0.1 + 0.05√3 + 0.3
= 0.4 + 0.05√3
= (8 + √3)/20 ≈ 0.487