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Mean and Variance of Continuous Distributions | 连续分布的均值与方差计算

2026-08-30 11:15 · 本文部分内容由AI辅助创作

📚 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：

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

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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：

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

9. Examination Tips for Edexcel | Edexcel 考试技巧