📚 IB Mathematics: Core Concepts and Exam Points of Continuous Random Variables | IB数学：连续随机变量核心概念与考点
Continuous random variables form a cornerstone of IB Mathematics Analysis and Approaches (AA) and Applications and Interpretation (AI) at both Standard Level (SL) and Higher Level (HL). This article systematically presents the essential definitions, properties, standard distributions, and examination strategies you need to master.
连续随机变量是IB数学分析与方法（AA）以及应用与解释（AI）标准级别（SL）和高级级别（HL）的核心内容。本文系统梳理了您需要掌握的基本定义、性质、常见分布以及考试策略。
1. Definition of Continuous Random Variables | 连续随机变量的定义
A continuous random variable X takes values over an interval or a union of intervals on the real number line. Unlike discrete variables, it can take any real value within a given range, and the probability of any single exact value is always zero.
连续随机变量X在实数轴上取一个区间或若干区间内的值。与离散变量不同，它在给定范围内可以取任意实数，且取任何单个精确值的概率始终为零。
For a continuous variable, P(X = a) = 0 for any specific value a.
Probabilities are evaluated over intervals: P(a ≤ X ≤ b), P(X < c), P(X > d), etc.
对于连续变量，任意特定值a都有P(X = a) = 0。
概率在区间上计算：P(a ≤ X ≤ b)、P(X < c)、P(X > d)等。
exclusion of endpoints does not change the probability: P(a ≤ X ≤ b) = P(a < X < b).
由于P(X = a) = 0，区间端点是否包含不影响概率值：P(a ≤ X ≤ b) = P(a < X < b)。
2. Probability Density Function (PDF) | 概率密度函数
A continuous random variable X is described by its probability density function f(x), which must satisfy two fundamental conditions for all real x.
连续随机变量X由其概率密度函数 f(x) 描述，该函数必须对所有实数x满足两个基本条件。
Condition 1: f(x) ≥ 0 for all x
条件1：对所有x，f(x) ≥ 0
Condition 2: ∫ f(x) dx from -∞ to +∞ = 1
条件2：从 -∞ 到 +∞ 的 ∫ f(x) dx = 1
The probability that X lies in the interval [a, b] is the area under the curve f(x) from a to b.
X落在区间 [a, b] 内的概率等于曲线 f(x) 在a到b之间的面积。
P(a ≤ X ≤ b) = ∫ f(x) dx from a to b
P(a ≤ X ≤ b) = 从a到b的 ∫ f(x) dx
3. Cumulative Distribution Function (CDF) | 累计分布函数
es a value less than or equal to x. It is the accumulated area under the PDF from -∞ to x.
累计分布函数 F(x) 给出了X取值小于或等于x的概率。它是概率密度函数从 -∞ 到 x 的累积面积。
F(x) = P(X ≤ x) = ∫ f(t) dt from -∞ to x
F(x) = P(X ≤ x) = 从 -∞ 到 x 的 ∫ f(t) dt
F'(x) = f(x), meaning the PDF is the derivative of the CDF.
P(a < X ≤ b) = F(b) − F(a).
As x → −∞, F(x) → 0; as x → +∞, F(x) → 1.
F'(x) = f(x)，即概率密度函数是累计分布函数的导数。
P(a < X ≤ b) = F(b) − F(a)。
当 x → −∞ 时，F(x) → 0；当 x → +∞ 时，F(x) → 1。
4. Expected Value (Mean) | 期望值（均值）
The expected value of a continuous random variable X, denoted E(X) or μ, represents the long-run average of X. It is calculated by integrating x multiplied by the PDF over the entire range.
连续随机变量X的期望值，记作 E(X) 或 μ，表示X的长期平均值。它通过在整个范围内对 x 乘以概率密度函数进行积分来计算。
E(X) = μ = ∫ x f(x) dx from -∞ to +∞
E(X) = μ = 从 -∞ 到 +∞ 的 ∫ x f(x) dx
or a function g(X) of the random variable, the expected value is E(g(X)) = ∫ g(x) f(x) dx.
对于随机变量的函数 g(X)，其期望值为 E(g(X)) = ∫ g(x) f(x) dx。
E(aX + b) = aE(X) + b (linearity property)
E(aX + b) = aE(X) + b（线性性质）
5. Variance and Standard Deviation | 方差与标准差
Variance measures the spread or dispersion of X around its mean μ. It is defined as the expected value of (X − μ)².
方差衡量X围绕其均值 μ 的离散程度。它定义为 (X − μ)² 的期望值。
Var(X) = σ² = E[(X − μ)²] = ∫ (x − μ)² f(x) dx
Var(X) = σ² = E[(X − μ)²] = ∫ (x − μ)² f(x) dx
An equivalent computational formula is often easier to use in examinations:
一个等价的简化计算公式在考试中往往更易使用：
Var(X) = E(X²) − [E(X)]²
Var(X) = E(X²) − [E(X)]²
Standard deviation: σ = √Var(X).
Var(aX + b) = a² Var(X), where a and b are constants.
标准差：σ = √Var(X)。
Var(aX + b) = a² Var(X)，其中a和b为常数。
6. Median and Quartiles | 中位数与四分位数
that splits the distribution into two equal halves. It satisfies the equation F(m) = 0.5.
连续随机变量X的中位数m是将分布分成两个相等一半的值。它满足方程 F(m) = 0.5。
∫ f(x) dx from -∞ to m = 0.5
从 -∞ 到 m 的 ∫ f(x) dx = 0.5
) = 0.75 respectively. These are obtained by solving the corresponding integral equations.
下四分位数 Q₁ 和上四分位数 Q₃ 分别满足 F(Q₁) = 0.25 和 F(Q₃) = 0.75。这些值通过求解相应的积分方程获得。
7. Mode of a Continuous Distribution | 连续分布的中位数
The mode of a continuous random variable is the value of x at which the probability density function f(x) attains its maximum. It represents the most likely value in a small neighbourhood.
连续随机变量的众数是概率密度函数 f(x) 达到最大值的x值。它表示在一个小邻域内最可能出现的值。
Find the critical points by solving f'(x) = 0.
Verify the maximum by checking f''(x) < 0 or by evaluating endpoints.
通过求解 f'(x) = 0 找到临界点。
通过检查 f''(x) < 0 或评估端点来验证最大值。
8. The Uniform Distribution | 均匀分布
A continuous uniform random variable X over the interval [a, b] has a constant probability density function. It is denoted X ~ U(a, b).
在区间 [a, b] 上的连续均匀随机变量X具有恒定的概率密度函数。记为 X ~ U(a, b)。
f(x) = 1 / (b − a) for a ≤ x ≤ b, and 0 otherwise
对于 a ≤ x ≤ b，f(x) = 1 / (b − a)；否则为0
| Property | Formula | 性质 | 公式 |
| CDF | F(x) = (x − a)/(b − a) | 累计分布函数 | F(x) = (x − a)/(b − a) |
| Mean | E(X) = (a + b)/2 | 均值 | E(X) = (a + b)/2 |
| Variance | Var(X) = (b − a)²/12 | 方差 | Var(X) = (b − a)²/12 |
9. The Normal Distribution | 正态分布
The normal distribution is the most important continuous distribution in IB Mathematics. A random variable X with mean μ and variance σ² is denoted X ~ N(μ, σ²). Its PDF is symmetric and bell-shaped.
正态分布是IB数学中最重要的连续分布。均值为 μ、方差为 σ² 的随机变量X记为 X ~ N(μ, σ²)。其概率密度函数对称且呈钟形。
f(x) = (1 / (σ √2π)) × e^(−(x − μ)² / (2σ²))
f(x) = (1 / (σ √2π)) × e^(−(x − μ)² / (2σ²))
The standard normal distribution is Z ~ N(0, 1). Any normal variable X can be standardised using the transformation:
标准正态分布为 Z ~ N(0, 1)。任何正态变量X都可以通过以下变换标准化：
Z = (X − μ) / σ
Z = (X − μ) / σ
IB exams expect you to use a GDC (graphical display calculator) to find normal probabilities.
Key symmetrical properties: P(Z < −z) = P(Z > z) and P(Z < z) = 1 − P(Z > z).
IB考试期望您使用图形计算器（GDC）来查找正态概率。
关键对称性质：P(Z < −z) = P(Z > z)，且 P(Z < z) = 1 − P(Z > z)。
10. Inverse Normal Distribution | 逆正态分布
ative probability. For example, finding the 90th percentile means solving P(X ≤ x) = 0.90.
逆正态问题要求找到与给定累计概率对应的x值。例如，查找第90百分位数意味着求解 P(X ≤ x) = 0.90。
Use the inverse normal function on your GDC: invNorm(p, μ, σ).
For the standard normal, use invNorm(p, 0, 1).
在GDC上使用逆正态函数：invNorm(p, μ, σ)。
对于标准正态分布，使用 invNorm(p, 0, 1)。
11. Worked Example | 例题精解
Problem: f(x) = 0 otherwise. Find: (a) the value of k; (b) P(0.5 ≤ X ≤ 1.5); (c) E(X); (d) Var(X).
题目：连续随机变量X的概率密度函数为 f(x) = k x（0 ≤ x ≤ 2），否则 f(x) = 0。求：(a) k的值；(b) P(0.5 ≤ X ≤ 1.5)；(c) E(X)；(d) Var(X)。
Solution (a): Since the total area must equal 1:
解 (a)：由于总面积必须等于1：
∫ kx dx from 0 to 2 = 1 → k[x²/2] from 0 to 2 = 1 → 2k = 1 → k = 0.5
从0到2的 ∫ kx dx = 1 → k[x²/2] 从0到2 = 1 → 2k = 1 → k = 0.5
Solution (b): X ≤ 1.5) = ∫ 0.5x dx from 0.5 to 1.5 = 0.25[x²] from 0.5 to 1.5 = 0.25(2.25 − 0.25) = 0.5.
解 (b)：P(0.5 ≤ X ≤ 1.5) = 从0.5到1.5的 ∫ 0.5x dx = 0.25[x²] 从0.5到1.5 = 0.25(2.25 − 0.25) = 0.5。
Solution (c): E(X) = ∫ x·0.5x dx from 0 to 2 = 0.5[x³/3] from 0 to 2 = 0.5(8/3) = 4/3 ≈ 1.333.
解 (c)：E(X) = 从0到2的 ∫ x·0.5x dx = 0.5[x³/3] 从0到2 = 0.5(8/3) = 4/3 ≈ 1.333。
Solution (d): to 2 = 0.5(16/4) = 2. Then Var(X) = E(X²) − [E(X)]² = 2 − (4/3)² = 2 − 16/9 = 2/9 ≈ 0.222.
解 (d)：首先，E(X²) = 0.5[x⁴/4] 从0到2 = 0.5(16/4) = 2。然后 Var(X) = E(X²) − [E(X)]² = 2 − (4/3)² = 2 − 16/9 = 2/9 ≈ 0.222。
12. Examination Tips and Common Pitfalls | 考试技巧与常见错误
Mastering the following strategies will help you avoid frequent mistakes in IB examinations.
掌握以下策略将帮助您在IB考试中避免常见错误。
| Tip | Explanation | 技巧 | 说明 |
| Always verify f(x) ≥ 0 | Check the domain and non-negativity before integration. | 始终验证 f(x) ≥ 0 | 在积分之前检查定义域和非负性。 |
| Use GDC efficiently | For normal distributions, set the mean and standard deviation correctly. | 高效使用GDC | 对于正态分布，正确设置均值和标准差。 |
| Don't confuse σ and σ² | In N(μ, σ²), the second parameter is the variance, not the standard deviation. | 不要混淆 σ 和 σ² | 在 N(μ, σ²) 中，第二个参数是方差，而不是标准差。 |
| State the formula | Write down integrals before evaluating to earn method marks. | 写出公式 | 在计算之前写出积分，以获得方法分。 |
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