Continuous Random Variables: Core Exam Points for IB Maths | IB数学连续随机变量核心考点

📚 Continuous Random Variables: Core Exam Points for IB Maths | IB数学连续随机变量核心考点

Continuous random variables are a fundamental topic in the IB Mathematics Analysis and Approaches (AA) and Applications and Interpretation (AI) syllabi. Understanding probability density functions, cumulative distribution functions, and their applications is essential for exam success.

连续随机变量是IB数学分析与方法（AA）以及应用与解读（AI）课程大纲中的核心专题。理解概率密度函数、累积分布函数及其应用，对考试取得高分至关重要。

1. Probability Density Function (PDF) | 概率密度函数

A continuous random variable X takes infinitely many possible values in an interval. Its behaviour is described by a probability density function f(x), which must satisfy two conditions:

连续随机变量 X 在一个区间内可取无穷多个值。它的行为由概率密度函数 f(x) 描述，该函数必须满足两个条件：

f(x) ≥ 0 for all x in the domain | 在定义域内，对所有 x 有 f(x) ≥ 0
∫ f(x) dx = 1 over the entire domain | 在整个定义域上，∫ f(x) dx = 1

The probability that X lies between a and b is the area under the curve: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx.

X 落在 a 和 b 之间的概率是曲线下的面积：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。

P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx

A common exam question asks you to find the value of a constant k such that f(x) is a valid PDF. Set the total integral equal to 1 and solve for k.

常见考题要求你找到常数 k 的值，使得 f(x) 是一个合法的概率密度函数。令总积分等于 1，然后解出 k。

2. Cumulative Distribution Function (CDF) | 累积分布函数

The cumulative distribution function F(x) gives the probability that X takes a value less than or equal to x:

累积分布函数 F(x) 给出 X 取值小于或等于 x 的概率：

F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt

Key properties of the CDF:

CDF 的关键性质：

0 ≤ F(x) ≤ 1 | 0 ≤ F(x) ≤ 1
F is non-decreasing | F 是单调不减的
limₓ→₋∞ F(x) = 0 and limₓ→∞ F(x) = 1 | limₓ→₋∞ F(x) = 0 且 limₓ→∞ F(x) = 1
P(a < X ≤ b) = F(b) − F(a) | P(a < X ≤ b) = F(b) − F(a)
f(x) = F′(x), where the derivative exists | f(x) = F′(x)，在导数存在处成立

To find the CDF from a piecewise PDF, integrate each piece and determine the constant of integration using continuity at the boundaries.

从分段 PDF 求 CDF 时，逐段积分，并利用边界处的连续性确定积分常数。

3. Expectation and Variance | 期望与方差

The mean or expected value of a continuous random variable is:

连续随机变量的均值或期望值为：

E(X) = ∫ x f(x) dx

The variance is defined as:

方差定义为：

Var(X) = E(X²) − [E(X)]² = ∫ x² f(x) dx − μ²

where μ = E(X). The standard deviation is σ = √Var(X).

其中 μ = E(X)。标准差为 σ = √Var(X)。

Exam tip: When computing E(X²), integrate x²·f(x), not (x·f(x))² — a common mistake students make.

考试提示：计算 E(X²) 时，是对 x²·f(x) 积分，而不是 (x·f(x))² —— 这是学生常犯的错误。

4. Median, Quartiles and Percentiles | 中位数、四分位数与百分位数

The median m of a continuous distribution satisfies:

连续分布的中位数 m 满足：

P(X ≤ m) = F(m) = 0.5

More generally, the p-th percentile xₚ satisfies F(xₚ) = p/100. The lower quartile Q₁ satisfies F(Q₁) = 0.25, and the upper quartile Q₃ satisfies F(Q₃) = 0.75.

更一般地，第 p 百分位数 xₚ 满足 F(xₚ) = p/100。下四分位数 Q₁ 满足 F(Q₁) = 0.25，上四分位数 Q₃ 满足 F(Q₃) = 0.75。

To solve such problems, set the CDF equal to the required probability and solve for x. If only the PDF is given, integrate first to obtain F(x) and then solve.

解此类问题时，令 CDF 等于所需概率，然后解出 x。如果题目只给出 PDF，先积分得到 F(x)，再求解。

A quick check: the median always lies between Q₁ and Q₃, and for a symmetric distribution, mean = median = mode.

快速检验：中位数总是位于 Q₁ 和 Q₃ 之间，且对于对称分布，均值 = 中位数 = 众数。

5. The Uniform Distribution | 均匀分布

The continuous uniform distribution on the interval [a, b] has PDF:

区间 [a, b] 上的连续均匀分布具有如下概率密度函数：

f(x) = 1/(b − a), for 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

The CDF is F(x) = (x − a)/(b − a) for a ≤ x ≤ b. These formulas are often tested directly in Paper 1, so memorise them carefully.

CDF 为 F(x) = (x − a)/(b − a)，其中 a ≤ x ≤ b。这些公式在Paper 1中经常直接考查，务必牢记。

6. The Normal Distribution | 正态分布

The normal distribution N(μ, σ²) is the most important continuous distribution in IB Maths. Its PDF is:

正态分布 N(μ, σ²) 是IB数学中最重要的连续分布。其概率密度函数为：

f(x) = (1/(σ√(2π))) · e^(−(x−μ)²/(2σ²))

Key facts about the normal distribution:

关于正态分布的关键事实：

It is symmetric about the mean μ | 关于均值 μ 对称
Approximately 68% of data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ | 约68%的数据落在均值1个标准差内，95%落在2个标准差内，99.7%落在3个标准差内
The standard normal variable Z = (X − μ)/σ has mean 0 and variance 1 | 标准正态变量 Z = (X − μ)/σ 的均值为0，方差为1

In IB exams, use your GDC to compute normal probabilities directly. For example, P(X < a) = normalCDF(−∞, a, μ, σ). The inverse normal function gives the x-value corresponding to a given cumulative probability.

在IB考试中，使用图形计算器（GDC）直接计算正态概率。例如，P(X < a) = normalCDF(−∞, a, μ, σ)。反正态函数给出与给定累积概率对应的 x 值。

Be careful: some questions specify X ~ N(μ, σ²) with variance given, while others give the standard deviation directly. Always check which one you are given.

注意：有些题目给出 X ~ N(μ, σ²) 时给的是方差，有些直接给标准差。务必确认题目给的是哪一个。

7. The Exponential Distribution | 指数分布

The exponential distribution with rate parameter λ has PDF:

参数为 λ 的指数分布具有如下概率密度函数：

f(x) = λe^(−λx), for x ≥ 0

Its CDF, mean and variance are:

其 CDF、均值和方差为：

F(x) = 1 − e^(−λx), E(X) = 1/λ, Var(X) = 1/λ²

The exponential distribution is often used to model waiting times and has the memoryless property: P(X > s + t | X > s) = P(X > t).

指数分布常用于建模等待时间，并具有无记忆性：P(X > s + t | X > s) = P(X > t)。

In exam questions, you may be asked to find λ from a given mean, compute probabilities using the CDF, or interpret the context of the problem (such as the time between arrivals).

在考题中，你可能会被要求从给定均值求出 λ，使用 CDF 计算概率，或解读问题的实际背景（例如到达间隔时间）。

8. Mode of a Continuous Distribution | 连续分布的众数

The mode of a continuous random variable is the value of x at which the PDF f(x) attains its maximum value.

连续随机变量的众数是使概率密度函数 f(x) 达到最大值的 x 值。

To find the mode:

求众数的方法：

Differentiate f(x) and solve f′(x) = 0 | 对 f(x) 求导，解方程 f′(x) = 0
Check the endpoints of the domain as well | 同时检查定义域的端点
Verify the critical point is a maximum using f″(x) or a sign table | 使用 f″(x) 或符号表验证临界点是极大值

For a symmetric distribution such as the normal distribution, the mode equals the mean and the median.

对于对称分布（如正态分布），众数等于均值和中位数。

9. Transformations of Random Variables | 随机变量的变换

If X is a continuous random variable and Y = aX + b, then:

如果 X 是连续随机变量，且 Y = aX + b，则：

E(Y) = aE(X) + b, Var(Y) = a²Var(X)

These linear transformation rules are frequently tested in connection with normal distributions. For example, if X ~ N(μ, σ²), then Y = aX + b ~ N(aμ + b, a²σ²).

这些线性变换规则经常与正态分布结合考查。例如，若 X ~ N(μ, σ²)，则 Y = aX + b ~ N(aμ + b, a²σ²)。

Note that adding a constant shifts the mean but does not affect the variance; multiplying by a constant scales both the mean and the standard deviation.

注意：加上常数会使均值平移但不影响方差；乘以常数会同时缩放均值和标准差。

10. Exam Strategies and Common Pitfalls | 考试策略与常见误区

Continuous random variable questions appear regularly in both IB Math AA and AI papers. Here are the key strategies:

连续随机变量题目在IB数学AA和AI试卷中经常出现。以下是关键策略：

Always verify the PDF integrates to 1 before proceeding | 解题前始终验证 PDF 的积分为 1
When finding constants, set up the integral equation first, then solve | 求常数时，先建立积分方程，再求解
Use the GDC efficiently: store functions, use the graphing feature to visualise, and check answers numerically | 高效使用 GDC：存储函数，利用绘图功能可视化，并数值验证答案
For normal distribution questions, sketch a bell curve and shade the required region | 对于正态分布题目，画出钟形曲线并标出所需区域
Distinguish between discrete and continuous cases: P(X = a) = 0 for continuous variables | 区分离散和连续情形：连续变量有 P(X = a) = 0

Common pitfalls include confusing the PDF with the CDF, forgetting that probabilities are areas (not function values), and misidentifying whether a given parameter is the variance or the standard deviation.

常见误区包括：混淆 PDF 与 CDF，忘记概率是面积（而不是函数值），以及未能区分题目给定的是方差还是标准差。

Mastering continuous random variables requires solid integration skills, a clear understanding of probability concepts, and confident use of your GDC. Practice past paper questions regularly and check every answer for mathematical reasonableness.

掌握连续随机变量需要扎实的积分技巧、清晰理解概率概念，以及熟练使用 GDC。定期练习历年真题，并检查每个答案在数学上是否合理。

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Continuous Random Variables: Core Exam Points for IB Maths | IB数学连续随机变量核心考点