Properties of Expected Value and Variance | 期望与方差的性质

📚 Properties of Expected Value and Variance | 期望与方差的性质

In probability and statistics, the expected value and variance are fundamental descriptors of a random variable's central tendency and spread. Their algebraic properties, such as linearity of expectation and the behaviour under transformation, greatly simplify complex probability calculations. This article explores the key properties of E(X) and Var(X) with clear explanations and examples relevant to IB Mathematics.

在概率与统计中，期望和方差是描述随机变量中心趋势和离散程度的基本量。它们的代数性质（如期望的线性性质和变换下方差的行为）极大地简化了复杂的概率计算。本文探究 E(X) 和 Var(X) 的关键性质，配以适合 IB 数学的清晰解释和示例。

1. Definition of Expectation E(X) | 期望的定义

For a discrete random variable X taking values x₁, x₂, ... with probabilities p₁, p₂, ..., the expected value is defined as E(X) = ∑ xᵗ p&# represents the long-run average of the outcomes if the experiment is repeated many times.

对于取值为 x₁, x₂, ... 且对应概率为 p₁, p₂, ... 的离散随机变量 X，期望定义为 E(X) = ∑ xᵗ pᵗ。它表示若实验重复多次时结果的长期平均值。

E(X) = ∑ xᵗ P(X = xᵗ)

inuous random variable with probability density function f(x), the expectation is E(X) = &# discrete and continuous variables, as they stem from the linearity of sums and integrals.

对于具有概率密度函数 f(x) 的连续随机变量，期望为 E(X) = ∫ x f(x) dx。本文讨论的性质对离散和连续变量都适用，因为它们源于求和与积分的线性性。

---

2. Linearity of Expectation | 期望的线性性质

his rule is a direct consequence of the linear nature of summation and integration, allowing constants to be factored out and constant additions to shift the mean directly.

期望满足 E(aX + b) = a E(X) + b，其中 a 和 b 为任意常数。该法则是求和与积分线性性质的直接结果，允许将常数提出，且常数加法直接平移均值。

E(aX + b) = a E(X) + b

at adding a constant simply shifts all outcomes by the same amount, while scaling the variable scales the expectation by the same factor.

作为特例，我们得到 E(X + c) = E(X) + c 及 E(cX) = c E(X)。这意味着加上一个常数只是将所有结果平移相同的量，而缩放变量则将期望缩放相同的倍数。

---

3. Expectation of a Sum of Random Variables | 随机变量和的期望

s: E(X + Y) = E(X) + E(Y). This property extends readily to the sum of multiple variables.

对于任意两个随机变量 X 和 Y，无论独立与否，它们和的期望等于各自期望的和：E(X + Y) = E(X) + E(Y)。这一性质很容易推广到多个变量的和。

E(X₁ + X₂ + ... + Xₙ) = E(X₁) + E(X₂) + ... + E(Xₙ)

e, the expected total score of two dice is simply 3.5 + 3.5 = 7, regardless of dependence.

这种可加性非常强大，因为它允许我们计算和的均值而无需变量的联合分布。例如，两个骰子总点数的期望就是 3.5 + 3.5 = 7，无论它们是否独立。

---

广告 / Sponsored

4. Expectation of a Constant and Constant Multiple | 常数的期望与常数倍期望

tant c is simply the constant itself: E(c) = c. This follows from the definition since the 'random variable' always takes the value c with probability 1, so the average is c.

常数 c 的期望就是常数本身：E(c) = c。这源自定义，因为该“随机变量”总以概率 1 取值 c，所以平均值为 c。

micro-properties form the building blocks of the general linearity rule described earlier.

结合线性性，对于任意常数 c 有 E(cX) = c E(X)。这些微性质构成了前述一般线性法则的基石。

E(c) = c,   E(cX) = c E(X)

---

5. Expectation of a Product of Independent Variables | 独立变量乘积的期望

es: E(XY) = E(X) E(Y). This relationship does not hold in general for dependent variables.

若 X 和 Y 是独立随机变量，则它们乘积的期望可分解：E(XY) = E(X) E(Y)。对于不独立的变量，此关系一般不成立。

If X and Y are independent, E(XY) = E(X) E(Y)

dicates zero covariance, but this does not guarantee full independence. Nevertheless, the product rule is a handy tool for simplifying expectations in problems involving independent events.

注意：逆命题不一定成立。E(XY) = E(X) E(Y) 意味着协方差为零，但这不保证完全独立。尽管如此，这条乘积规则是简化涉及独立事件期望问题的便捷工具。

---

6. Definition of Variance Var(X) | 方差的定义

Variance measures how spread out the values of a random variable are around its mean μ = E(X). It is defined as the expected squared deviation: Var(X) = E[(X − μ)²]. The units of variance are the square of the units of X.

方差衡量随机变量值围绕其均值 μ = E(X) 的离散程度。它定义为偏差平方的期望：Var(X) = E[(X − μ)²]。方差的单位是 X 单位的平方。

Var(X) = E[(X − μ)²]

computationally more convenient formula is obtained by expanding the square: Var(X) = E(X²) − [E(X)]&# form is often used to calculate variance directly from the probability distribution of X.

展开平方可得一个计算上更方便的公式：Var(X) = E(X²) − [E(X)]²。这一形式常用于直接从 X 的概率分布计算方差。

Var(X) = E(X²) − [E(X)]²

---

7. Variance of a Linear Transformation | 线性变换的方差

any constants a and b, the variance of the transformed variable aX + b is Var(aX + b) = a&# affect the variance because it merely shifts the distribution without changing its spread.

对于任意常数 a 和 b，变换后变量 aX + b 的方差为 Var(aX + b) = a² Var(X)。注意加上常数 b 不影响方差，因为它仅仅平移分布而不改变离散程度。

Var(aX + b) = a² Var(X)

As particular cases, Var(X + b) = Var(X) and Var(aX) = a&#; Var(X). Multiplying a variable by a negative factor a still increases the variance by a² because variance is based on squared deviations.

作为特例

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）