📚 Mean and Variance of Continuous Distributions | 连续分布的均值与方差计算
In A-Level Further Mathematics, discrete distributions such as the binomial or Poisson distribution deal with outcomes that take countable values. However, many real-world quantities — waiting times, measurement errors, physical dimensions — vary continuously. Understanding how to compute the mean and variance of a continuous random variable is therefore essential for modelling problems in mechanics, statistics and finance.
在 A-Level 进阶数学中，二项分布和泊松分布等离散分布处理的是可数结果。然而，许多现实量——等待时间、测量误差、物理尺寸——是连续变化的。因此，理解如何计算连续随机变量的均值与方差，对于在力学、统计和金融中建模问题至关重要。
1. The Probability Density Function | 概率密度函数
A continuous random variable X is described by a probability density function (PDF) f(x), which satisfies two key conditions:
连续随机变量 X 由概率密度函数 (PDF) f(x) 描述，它满足两个关键条件：
- f(x) ≥ 0 for all x in the sample space.
- 对所有 x，f(x) ≥ 0。
- The total area under the curve equals 1: ∫ f(x) dx = 1 over the full range.
- 曲线下的总面积为 1： 在整个范围内 ∫ f(x) dx = 1。
Probabilities are found by integrating the PDF over an interval:
概率通过对 PDF 在区间上积分求得：
P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx
2. Definition of the Mean of a Continuous Distribution | 连续分布均值的定义
For a discrete distribution, the mean is E(X) = Σ x P(X = x). For a continuous distribution, the summation is replaced by integration:
对于离散分布，均值为 E(X) = Σ x P(X = x)。对于连续分布，求和被积分替代：
E(X) = μ = ∫ x f(x) dx
where the integral is taken over the entire range of X. The mean is also called the expectation or the first moment of the distribution.
其中积分在 X 的整个取值范围内进行。均值也称为期望或分布的一阶矩。
3. Definition of the Variance and Standard Deviation | 方差与标准差定义
The variance measures the spread of the distribution around the mean. It is defined as:
方差衡量分布围绕均值的离散程度。其定义为：
Var(X) = E[(X − μ)²] = ∫ (x − μ)² f(x) dx
An equivalent and often simpler formula is:
一个等价且常常更简便的公式是：
Var(X) = E(X²) − [E(X)]²
where E(X²) = ∫ x² f(x) dx. The standard deviation is the positive square root of the variance, and shares the same units as X.
其中 E(X²) = ∫ x² f(x) dx。标准差是方差的正平方根，与 X 具有相同的单位。
4. Worked Example 1: A Linear PDF | 例 1：线性概率密度函数
Example: A continuous random variable X has PDF f(x) = kx for 0 ≤ x ≤ 2, and 0 otherwise. Find the value of k, then compute E(X) and Var(X).
示例： 连续随机变量 X 的概率密度函数为 0 ≤ x ≤ 2 时 f(x) = kx，否则为 0。求 k 的值，然后计算 E(X) 和 Var(X)。
Step 1 — Normalisation:
步骤 1 — 归一化：
∫₀² kx dx = 1 → k[x²/2]₀² = 1 → 2k = 1 → k = ½
Step 2 — Mean:
步骤 2 — 均值：
E(X) = ∫₀² x · (½x) dx = ∫₀² ½x² dx = [x³/6]₀² = 8/6 = 4/3
Step 3 — Second moment:
步骤 3 — 二阶矩：
E(X²) = ∫₀² x² · (½x) dx = ∫₀² ½x³ dx = [x⁴/8]₀² = 16/8 = 2
Step 4 — Variance:
步骤 4 — 方差：
Var(X) = 2 − (4/3)² = 2 − 16/9 = 2/9
5. Worked Example 2: A Quadratic PDF | 例 2：二次概率密度函数
Example: The random variable Y has PDF f(y) = 3y² for 0 ≤ y ≤ 1. Find E(Y) and Var(Y).
示例： 随机变量 Y 的概率密度函数为 0 ≤ y ≤ 1 时 f(y) = 3y²。求 E(Y) 和 Var(Y)。
First note that the total integral is already 1, since ∫₀¹ 3y² dy = [y³]₀¹ = 1.
首先注意总积分已经为 1，因为 ∫₀¹ 3y² dy = [y³]₀¹ = 1。
E(Y) = ∫₀¹ y · 3y² dy = ∫₀¹ 3y³ dy = [3y⁴/4]₀¹ = 3/4
E(Y²) = ∫₀¹ y² · 3y² dy = ∫₀¹ 3y⁴ dy = [3y⁵/5]₀¹ = 3/5
Var(Y) = 3/5 − (3/4)² = 3/5 − 9/16 = (48 − 45)/80 = 3/80
6. The Mode, Median and the Mean | 众数、中位数与均值
Beyond the mean, two other measures of central tendency are commonly examined:
除均值外，还有两个常见的集中趋势度量：
- Mode: the value of x at which f(x) attains its maximum. For a differentiable PDF, solve f‘(x) = 0 and verify the nature of the stationary point.
- 众数： f(x) 达到最大值时的 x 值。对于可微的 PDF，解 f‘(x) = 0 并验证驻点的性质。
- Median: the value m such that P(X ≤ m) = ½, i.e. ∫₋∞ᵐ f(x) dx = ½.
- 中位数： 满足 P(X ≤ m) = ½ 的值 m，即 ∫₋∞ᵐ f(x) dx = ½。
For a symmetric distribution, the mean, median and mode coincide. For skewed distributions, they are generally different, and their ordering depends on the direction of the skew.
对于对称分布，均值、中位数和众数重合。对于偏态分布，它们通常不同，其排列顺序取决于偏斜的方向。
7. The Uniform Distribution | 均匀分布
The continuous uniform distribution on the interval [a, b] has PDF:
区间 [a, b] 上的连续均匀分布的 PDF 为：
f(x) = 1/(b − a), a ≤ x ≤ b
Its mean and variance are:
其均值和方差为：
E(X) = (a + b)/2, Var(X) = (b − a)²/12
For example, if X ~ U(2, 6), then E(X) = 4 and Var(X) = 16/12 = 4/3.
例如，如果 X ~ U(2, 6)，则 E(X) = 4，Var(X) = 16/12 = 4/3。
8. Common Mistakes Students Make | 学生常犯的错误
- Forgetting to normalise the PDF: always check ∫ f(x) dx = 1 first, especially when a constant k is present.
- 忘记归一化 PDF： 首先务必检查 ∫ f(x) dx = 1，尤其是当存在常数 k 时。
- Confusing E(X²) with [E(X)]²: the variance is the difference between these two quantities, not the ratio or the square of the mean alone.
- 将 E(X²) 与 [E(X)]² 混淆： 方差是这两个量之差，而非比值，也不仅仅是均值的平方。
- Integrating over the wrong limits: apply the limits for which the PDF is non-zero, not over an arbitrary range.
- 积分限用错： 应使用 PDF 非零的区间作为积分限，而非任意范围。
- Using discrete formulas: do not attempt to sum probabilities; integration is required for continuous distributions.
- 使用离散公式： 不要尝试对概率求和；连续分布必须使用积分。
9. Examination Tips for Edexcel | Edexcel 考试技巧
- Always state the formula before substituting values. Edexcel mark schemes award method marks for correct formulas even if the final answer is wrong.
- 在代入数值之前始终写出公式。Edexcel 评分方案会对正确的公式给方法分，即使最终答案错误。
- Present your working in the order: normalise → E(X) → E
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