📚 9B Variance of a discrete random variable | 9B 离散随机变量的方差
Understanding variance is crucial when working with random variables in IB Mathematics. While the expected value tells us the average outcome, variance measures the spread or variability of the outcomes around this average. In this article, we explore the definition, formulas, and properties of variance for a discrete random variable, as covered in section 9B of the IB syllabus. We'll work through examples and discuss how to avoid common mistakes.
在 IB 数学中处理随机变量时，理解方差至关重要。期望值告诉了我们平均结果，而方差衡量的是结果围绕该平均值的分散程度或变异性。在本文中，我们将探讨 IB 教学大纲第 9B 节中离散随机变量的方差的定义、公式和性质。我们将通过实例进行讲解，并讨论如何避免常见错误。
1. Expected Value: A Quick Review | 期望值：快速回顾
(X = xᵢ). This weighted average acts as the balance point of the probability distribution.
在深入探讨方差之前，请回顾：对于一个离散随机变量 X，其可能取值为 x₁, x₂, ..., xₙ，对应概率为 P(X = xᵢ)，期望值（均值）为 E(X) = μ = Σ xᵢ · P(X = xᵢ)。这个加权平均值是概率分布的平衡点。
E(X) = μ = Σ xᵢ · P(X = xᵢ)
E(X) = μ = Σ xᵢ · P(X = xᵢ)
For example, if X represents the score on a biased die with probabilities, we first compute μ before evaluating variance. Always verify that the sum of all probabilities equals 1.
例如，若 X 表示一个有偏骰子的得分且带有相应概率，我们需要先计算 μ 再计算方差。务必先验证所有概率之和等于 1。
2. Introducing Variance | 方差的引入
The expected value alone does not tell us how reliable that average is. Two different distributions can share the same mean yet spread their probabilities very differently. Variance quantifies dispersion by averaging the squared deviations from the mean, giving greater weight to larger deviations.
仅凭期望值并不能告诉我们该平均值的可靠程度。两个不同的分布可以拥有相同的均值，但概率的散布方式却截然不同。方差通过平均与均值的偏差的平方来量化离散程度，对较大的偏差给予更大的权重。
Without a measure of spread, we cannot fully understand the risk or uncertainty associated with a random variable. Higher variance implies outcomes are more widely scattered around μ.
如果没有对分散程度的度量，我们就无法完全理解与随机变量相关的风险或不确定性。方差越大，意味着结果在 μ 周围的分散程度越广。
3. Definition and Formula | 定义与公式
The variance of a discrete random variable X is denoted by Var(X) or σ². It is defined as the expected value of the squared deviation from the mean:
离散随机变量 X 的方差记作 Var(X) 或 σ²，其定义为与均值偏差的平方的期望值：
Var(X) = E[(X - μ)²] = Σ (xᵢ - μ)² · P(X = xᵢ)
Var(X) = E[(X - μ)²] = Σ (xᵢ - μ)² · P(X = xᵢ)
In this expression, each possible value xᵢ has its deviation from μ squared and multiplied by its probability. The sum of these weighted squared deviations gives the variance.
在该表达式中，每个可能的取值 xᵢ 先计算它与 μ 的偏差，然后平方，再乘以其概率。这些加权平方偏差的总和就是方差。
ility mass is concentrated at a single value, then every deviation is zero, so Var(X) = 0.
由于我们对偏差进行了平方，方差始终是非负的。如果所有概率集中在一个单一值上，那么每个偏差都为零，因此 Var(X) = 0。
4. The Shortcut Formula (Computational Form) | 速算公式（计算形式）
In practice, computing (xᵢ - μ)² for each value can be tedious. A more efficient method is the computational formula:
在实践中，对每个值计算 (xᵢ - μ)² 可能很繁琐。一种更高效的方法是计算形式的公式：
Var(X) = E(X²) - [E(X)]² = Σ xᵢ² · P(X = xᵢ) - μ²
Var(X) = E(X²) - [E(X)]² = Σ xᵢ² · P(X = xᵢ) - μ²
cut avoids subtracting the mean from each observation, which is especially useful when the mean is not an integer.
这里，E(X²) 是随机变量平方的期望值，通过对 xᵢ² · P(X = xᵢ) 求和得到。这个速算方法避免了从每个观测值中减去均值，当均值不是整数时尤其有用。
= E(X²) - μ², using linearity of expectation. This identity is a key tool in the IB exam.
要推导该公式，可将 E[(X - μ)²] 展开为 E(X² - 2μX + μ²) = E(X²) - 2μ E(X) + μ² = E(X²) - 2μ² + μ² = E(X²) - μ²，其中运用了期望的线性性质。这个恒等式是 IB 考试中的一项重要工具。
5. Variance of a Linear Transformation | 线性变换的方差
When we apply a linear transformation to a random variable, the variance behaves in a predictable way. For constants a and b:
当我们对随机变量进行线性变换时，方差的变化方式是可预测的。对于常数 a 和 b：
Var(aX + b) = a² Var(X)
Var(aX + b) = a² Var(X)
Adding the constant b shifts every value by the same amount, so deviations from the mean remain unchanged, and variance is unaffected. Multiplying by a scales all values, making deviations a times larger, so their squares become a² times larger.
加上常数 b 会将每个值平移相同的量，因此与均值的偏差保持不变，方差不受影响。乘以 a 会缩放所有值，使得偏差变为原来的 a 倍，因此偏差的平方变为 a² 倍。
ty is frequently tested in IB questions involving transformations of probability distributions.
例如，若 X 的方差为 4，则 Var(3X - 2) = 3² · 4 = 36，而 Var(-X) = (-1)² Var(X) = Var(X)。在涉及概率分布变换的 IB 题目中，这一性质经常被考查。
6. Standard Deviation | 标准差
r(X), brings the measure back to the original units and is widely used to describe spread.
由于方差的单位是 X 的平方单位，直接解释它可能很别扭。标准差 σ = √Var(X) 将度量拉回到原单位，被广泛用于描述离散程度。
In IB problems, you may be asked for either the variance or the standard deviation. Read the question carefully: 'calculate the variance' versus 'calculate the standard deviation'. Always state the formula before substituting numbers to secure method marks.
在 IB 试题中，可能要求你计算方差或标准差。请仔细读题：“计算方差”与“计算标准差”是不同的。在代入数字前，务必先写出公式，以确保获得方法分。
7. Worked Example: Calculating Variance Step by Step | 实例：逐步计算方差
Consider a discrete random variable X with the following probability distribution:
考虑一个离散随机变量 X，其概率分布如下：
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| P(X=x) | 0.2 | 0.3 | 0.4 | 0.1 |
First, compute the expected value μ:
首先，计算期望值 μ：
μ = (1 × 0.2) + (2 × 0.3) + (3 × 0.4) + (4 × 0.1) = 0.2 + 0.6 + 1.2 + 0.4 = 2.4
μ = (1 × 0.2) + (2 × 0.3) + (3 × 0.4) + (4 × 0.1) = 0.2 + 0.6 + 1.2 + 0.4 = 2.4
the definition Var(X) = Σ (x - μ)² P(X=x). Construct a table to organise the calculations:
现在应用定义 Var(X) = Σ (x - μ)² P(X=x)。构建一个表格来整理计算过程：
| x | P(X=x) | x - μ | (x - μ)² | (x - μ)² P(X=x) |
|---|---|---|---|---|
| 1 | 0.2 | -1.4 | 1.96 | 0.392 |
| 2 | 0.3 | -0.4 | 0.16 | 0.048 |
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