📚 IB Mathematics: Introduction to Continuous Random Variables | IB数学：连续随机变量概论
In IB Mathematics (Analysis and Approaches or Applications and Interpretation), continuous random variables form a foundational topic in probability and statistics. Unlike discrete random variables, which take countable values, continuous random variables take values on an interval — infinite possibilities between any two points. This article introduces the key concepts, formulas, and exam strategies you need to master this topic.
在IB数学（分析与方法或应用与解释）课程中，连续随机变量是概率与统计部分的基础内容。与取可数值的离散随机变量不同，连续随机变量在某个区间上取值，任意两点之间都有无限多个可能值。本文将介绍你需要掌握的核心概念、公式与考试技巧。
1. Definition of Continuous Random Variables | 连续随机变量的定义
A continuous random variable X is one that can take any real value within a given range or interval. In practice, such variables arise from measurements: time, length, temperature, speed, or voltage. For example, the time a student spends completing a test is a continuous random variable because it can be any positive number, not just whole numbers.
连续随机变量 X 是指在给定区间内可以取任意实数值的随机变量。实践中，这类变量来自测量：时间、长度、温度、速度或电压。例如，学生完成一份测试所花的时间就是连续随机变量，因为它可以取任意正数，而不只是整数。
Because there are infinitely many possible values, the probability that X takes an exact single value is zero: P(X = c) = 0 for any constant c. Probabilities are instead associated with intervals, such as P(a ≤ X ≤ b).
由于可能取值有无限多个，X 恰好取某个特定值的概率为零：对任意常数 c，P(X = c) = 0。概率因此与区间相关，例如 P(a ≤ X ≤ b)。
2. Probability Density Function (PDF) | 概率密度函数
The probability distribution of a continuous random variable is described by a probability density function, f(x). The PDF is not a probability; rather, the area under the curve f(x) between two limits gives the probability of an interval.
连续随机变量的概率分布由概率密度函数 f(x) 描述。PDF 本身不是概率，而是曲线 f(x) 在区间下方围成的面积给出该区间的概率。
P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx
For example, if f(x) represents the density of waiting times at a bus stop, then the area under f(x) from 2 minutes to 5 minutes equals the probability that wait time falls in that interval.
例如，如果 f(x) 表示公交站等车时间的密度，那么 f(x) 在 2 分钟到 5 分钟之间围成的面积就等于等车时间落在此区间内的概率。
3. Properties of the PDF | 概率密度函数的性质
Every valid probability density function must satisfy two essential conditions. First, f(x) must be non-negative for all x — a negative density would produce negative probabilities, which is meaningless.
任何有效的概率密度函数都必须满足两个基本条件。第一，对所有 x，f(x) 必须非负——负密度会导致负概率，这没有意义。
Second, the total area under the curve must equal 1, representing the certain event that X takes some value in its domain.
第二，曲线下方的总面积必须等于 1，表示 X 在其定义域内取某个值这一必然事件。
∫₋∞⁺∞ f(x) dx = 1
When solving problems, the first step is almost always to find an unknown constant k in the PDF by applying this normalisation condition.
解题时的第一步几乎总是利用这个归一化条件来确定 PDF 中的未知常数 k。
4. Cumulative Distribution Function (CDF) | 累积分布函数
The cumulative distribution function, F(x), gives the probability that X is less than or equal to a particular value x. It accumulates all the probability density from −∞ up to x.
累积分布函数 F(x) 给出 X 小于或等于某个特定值 x 的概率。它将从 −∞ 到 x 的所有概率密度累积起来。
F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt
The CDF has three key properties: it is non-decreasing; it approaches 0 as x → −∞; and it approaches 1 as x → +∞. In IB exams, you may be asked to interpret F(x) directly or to use it to find probabilities such as P(X > a) = 1 − F(a).
CDF 有三个关键性质：非递减；当 x → −∞ 时趋近于 0；当 x → +∞ 时趋近于 1。在IB考试中，你可能被要求直接解读 F(x)，或利用它求概率，例如 P(X > a) = 1 − F(a)。
5. Relationship between PDF and CDF | PDF与CDF的关系
The PDF and CDF are closely connected by differentiation and integration. In regions where f(x) is continuous, the derivative of the CDF is exactly the PDF.
PDF 与 CDF 通过微分与积分紧密相连。在 f(x) 连续的区域，CDF 的导数恰为 PDF。
f(x) = F ‘(x)
This relationship allows you to move between the two functions in either direction. If a question provides F(x), you can find f(x) by differentiating; if it provides f(x), you can find F(x) by integrating. This is a classic IB short-response question.
这一关系允许你在两个函数之间双向转换。若题目给出 F(x)，可通过求导得到 f(x)；若给出 f(x)，可通过积分得到 F(x)。这是经典的IB简答题。
6. Expected Value (Mean) | 期望值（均值）
The expected value of a continuous random variable is analogous to the mean of a distribution. It is calculated by integrating x multiplied by the PDF over the entire domain.
连续随机变量的期望值类似于分布的平均值。计算方法是在整个定义域内对 x 乘以 PDF 求积分。
E[X] = ∫₋∞⁺∞ x f(x) dx
Conceptually, E[X] represents the long-run average value of X if the random experiment is repeated infinitely many times. For symmetric PDFs, the mean lies at the centre of symmetry; for skewed PDFs, the mean shifts toward the longer tail.
概念上，E[X] 表示若随机试验无限次重复，X 的长期平均值。对于对称的 PDF，均值位于对称中心；对于偏态 PDF，均值向较长尾部方向偏移。
7. Variance and Standard Deviation | 方差与标准差
Variance measures the spread of a continuous random variable around its mean. It is defined as the expected squared deviation from the mean, and can be computed using the formula:
方差衡量连续随机变量在其均值周围的分散程度。它定义为对均值偏差平方的期望，可通过以下公式计算：
Var(X) = E[(X − μ)²] = E[X²] − μ²
where μ = E[X] and E[X²] = ∫₋∞⁺∞ x² f(x) dx. The standard deviation is simply the positive square root of the variance: σ = √Var(X). In IB, you must be comfortable computing both E[X] and E[X²] to obtain variance.
其中 μ = E[X]，且 E[X²] = ∫₋∞⁺∞ x² f(x) dx。标准差就是方差的正平方根：σ = √Var(X)。在IB中，你必须熟练掌握计算 E[X] 和 E[X²] 来求方差。
8. Mode and Median | 众数和中位数
The mode of a continuous random variable is the value of x at which the PDF f(x) attains its maximum. It is found by differentiating f(x) and setting the derivative to zero, or by inspecting the graph if the mode is at an endpoint.
连续随机变量的众数是使 PDF f(x) 达到最大值的 x 值。求法是对 f(x) 求导并令导数为零，或如果众数在端点处，则直接观察图像。
The median m is the value that splits the distribution into two equal halves: half the probability lies below m and half above. It satisfies the equation F(m) = 0.5, or equivalently ∫₋∞ᵐ f(x) dx = 0.5.
中位数 m 是将分布分为两等份的值：一半概率在 m 之下，一半在 m 之上。它满足方程 F(m) = 0.5，等价于 ∫₋∞ᵐ f(x) dx = 0.5。
For a perfectly symmetric distribution, the mean, median, and mode coincide. For skewed distributions, they differ in a predictable order, which is a useful qualitative check in exam problems.
对于完全对称的分布，均值、中位数和众数重合。对于偏态分布，它们按可预测的顺序排列，这在考试解题中非常实用的定性检验。
9. Continuous Uniform Distribution | 连续均匀分布
The simplest continuous distribution is the uniform distribution on the interval [a, b]. Its PDF is constant over this interval, meaning every subinterval of equal length has equal probability.
最简单的连续分布是区间 [a, b] 上的均匀分布。其 PDF 在该区间内为常数，意味着长度相等的任意子区间有相等的概率。
f(x) = 1 ⁄ (b − a) for a ≤ x ≤ b, zero otherwise
For a uniform random variable X ~ U(a, b), the mean and variance have the closed forms:
对于均匀随机变量 X ~ U(a, b)，其均值和方差有闭合形式：
E[X] = (a + b) ⁄ 2, Var(X) = (b − a)² ⁄ 12
These formulas are quick to apply and appear frequently in IB multiple-choice and short-answer items. They also serve as building blocks for simulating other continuous distributions using technology.
这些公式应用快捷，在IB选择题和简答题中经常出现。它们也是使用技术工具模拟其他连续分布的基础模块。
10. Worked Example | 例题讲解
Let X be a continuous random variable with PDF f(x) = kx for 0 ≤ x ≤ 2, and f(x) = 0 otherwise. Find the value of k, the probability P(0.5 ≤ X ≤ 1.5), the expected value E[X], and the variance Var(X).
设 X 为连续随机变量，其 PDF 为 f(x) = kx（0 ≤ x ≤ 2），其余为 f(x) = 0。求常数 k 的值、概率 P(0.5 ≤ X ≤ 1.5)、期望值 E[X] 和方差 Var(X)。
Step 1: Use the normalisation condition. Integrate f(x) from 0 to 2 and set the result equal to 1:
第一步：使用归一化条件。将 f(x) 从 0 到 2 积分并令结果等于 1：
∫₀² kx dx = k · [x²/2]₀² = k · 2 = 1 ⇒ k = 1/2 = 0.5
Step 2: Find the interval probability by integrating the PDF from 0.5 to 1.5:
第二步：通过从 0.5 到 1.5 积分 PDF 求区间概率：
P(0.5 ≤ X ≤ 1.5) = ∫₀.₅¹·⁵ 0.5x dx = 0.25[x²]₀.₅¹·⁵ = 0.25(2.25 − 0.25) = 0.5
Step 3: Compute E[X] using the definition:
第三步：运用定义计算 E[X]：
E[X] = ∫₀² x · 0.5x dx = ∫₀² 0.5x² dx = 0.5 · [x³/3]₀² = 0.5 · (8/3) = 4/3
Step 4: Compute E[X²] and then the variance:
第四步：计算 E[X²] 然后计算方差：
E[X²] = ∫₀² x² · 0.5x dx = 0.5 · [x⁴/4]₀² = 0.5 · 4 = 2
Var(X) = E[X²] − (E[X])² = 2 − (16/9) = 2/9
This example illustrates the full process: normalise to find constants, integrate over the interval for probabilities, and use E[X²] to obtain variance.
此例展示了完整过程：归一化求常数、对区间积分求概率、利用 E[X²] 求方差。
11. Common Pitfalls | 常见易错点
One common error is treating f(x) as a probability. For example, f(1) = 0.5 does not mean P(X = 1) = 0.5; in fact, P(X = 1) = 0. Instead, you must integrate f(x) over an interval to get a probability.
常见错误之一是将 f(x) 当作概率。例如，f(1) = 0.5 并不意味着 P(X = 1) = 0.5；实际上，P(X = 1) = 0。相反，你必须对 f(x) 在某个区间上积分才能得到概率。
Another frequent mistake is forgetting to check the normalisation condition before using a PDF. If an unknown constant is present, always solve ∫f(x)dx = 1 first. Additionally, be careful with integration limits: outside the defined domain, f(x) = 0, so you must not integrate beyond the endpoints.
另一个常见错误是在使用 PDF 前忘记检查归一化条件。如果存在未知常数，务必先求解 ∫f(x)dx = 1。此外，要小心积分上下限：在定义域之外 f(x) = 0，因此不能超出端点积分。
A third pitfall is using the discrete variance formula directly. For continuous variables, always use Var(X) = E[X²] − (E[X])² with E[X²] found via integration, not summation.
第三个易错点是直接套用离散方差公式。对于连续变量，务必通过积分（而非求和）得到 E[X²]，再使用 Var(X) = E[X²] − (E[X])²。
12. Exam Tips | 考试技巧
In IB exams, you may use a GDC to evaluate definite integrals for probabilities and expectations, but you must still show the correct integral setup in your working. Write the integral expression clearly before using technology.
在IB考试中，你可以使用图形计算器（GDC）计算定积分来求概率和期望值，但必须在解答过程中正确写出积分表达式。使用计算器之前，请清晰写出积分式子。
Sketch the PDF whenever possible. Even a rough graph helps you confirm whether your answer is reasonable: probabilities correspond to areas, and the total area must be 1. For symmetric distributions, the mean and median coincide—use this as a quick check.
尽可能画出 PDF 的图像。即使是粗糙的草图也能帮助你确认答案是否合理：概率对应面积，总面积必须为 1。对于对称分布，均值和中位数重合——可用来快速检验。
Finally, memorise the uniform distribution formulas and the fundamental relationships F'(x) = f(x) and E[X] = ∫x f(x)dx. These patterns appear repeatedly on IB paper 1 and paper 2, and mastering them will save valuable time.
最后，熟记均匀分布公式以及基本关系 F'(x) = f(x) 和 E[X] = ∫x f(x)dx。这些模式在IB卷一和卷二反复出现，掌握它们能节省宝贵时间。
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