📚 Mastering Probability Density Functions for Edexcel A-Level Maths | 掌握Edexcel A-Level数学中的概率密度函数
In Edexcel A-Level Mathematics, particularly in the Statistics modules (Year 2), probability density functions (PDFs) play a central role in modelling continuous random variables. Understanding how to work with PDFs is essential for tackling questions on continuous distributions, whether it is finding probabilities, calculating measures of central tendency, or deriving the cumulative distribution function. This comprehensive revision guide breaks down every key concept you need, from definitions and validation to expectation, variance, and exam-style problem solving.
在Edexcel A-Level数学课程中，特别是在统计模块（第二年）中，概率密度函数（PDF）在连续型随机变量建模中扮演着核心角色。无论是求概率、计算集中趋势度量，还是推导累积分布函数，掌握PDF的运算方法都是解题的关键。这份全面复习指南将逐一拆解你需要的每一个核心概念，从定义与验证，到期望、方差以及考试题型实战。
1. Continuous Random Variables – The Big Picture | 连续随机变量——整体框架
points but to intervals. The tool that describes the probability distribution of a continuous random variable is the probability density function, often denoted by f(x). It is important to remember that f(x) itself does not give a probability directly; the probability that X lies in an interval [a, b] is given by the area under the curve of f(x) between a and b, that is, P(a ≤ X ≤ b) = ∫ab f(x) dx.
连续随机变量可以在给定区间内取任何值。与离散变量不同，概率并非分配给单个点，而是分配给区间。描述连续随机变量概率分布的工具就是概率密度函数，通常记作 f(x)。必须牢记，f(x) 本身并不直接给出概率；随机变量 X 落在区间 [a, b] 内的概率由曲线 f(x) 下方介于 a 与 b 之间的面积给出，即 P(a ≤ X ≤ b) = ∫ab f(x) dx。
2. Defining a Probability Density Function | 概率密度函数定义
) ≥ 0 for all x in R, and the total area under the curve over the entire range equals 1, written as ∫R for example f(x) = kx for 0 ≤ x ≤ 2, and zero otherwise, and asked to find the constant k.
如果一个函数 f(x) 在其定义域 R 上满足两个条件，那么它就是一个有效的概率密度函数：对于 R 内的所有 x 有 f(x) ≥ 0，并且在整个定义域上曲线下的总面积为 1，即 ∫R f(x) dx = 1。在定义域之外，我们通常令 f(x) = 0。在Edexcel考题中，你常常会碰到分段定义的函数，例如 f(x) = kx，0 ≤ x ≤ 2，其他地方为零，然后要求你求出常数 k。
3. Validating a PDF and Finding Unknown Constants | 验证PDF与求未知常数
The most common small mark question involves using the total area property to determine an unknown coefficient. You simply integrate f(x) over its domain, set the result equal to 1, and solve for the constant. Always check the non-negativity condition: after finding k, verify that f(x) does not become negative anywhere in the given interval. If the domain consists of multiple intervals, you must integrate piece by piece and sum the integrals.
最常见的小分值问题就是利用总面积性质来确定未知系数。你只需对 f(x) 在其定义域上积分，令结果等于 1，然后解出常数即可。一定要检查非负条件：求出 k 之后，确认 f(x) 在给定区间内没有变成负值。如果定义域由多个区间组成，你必须逐段积分并将积分值相加。
- Example: For f(x) = k(4 – x²) on 0 ≤ x ≤ 2, find k. | 示例：已知 f(x) = k(4 – x²)，0 ≤ x ≤ 2，求 k。
- Compute: ∫02 k(4 – x²) dx = k [4x – x³/3]₂₀ = k(8 – 8/3) = (16/3)k = 1 → k = 3/16. | 计算：∫02 k(4 – x²) dx = k [4x – x³/3]₀² = k(8 – 8/3) = (16/3)k = 1 → k = 3/16。
4. Cumulative Distribution Function (CDF) | 累积分布函数
with PDF f(x), the cumulative distribution function F(x) is defined as F(x) = P(X ≤ x) = ∫-∞x f(t) dt. The CDF is a non-decreasing function that rises from 0 to 1 as x increases. In practice, you integrate the PDF from the lower boundary of the domain up to x. Edexcel questions frequently ask you to specify F(x) for different ranges of x, typically resulting in a piecewise definition that includes 0 for values below the domain and 1 for values above the domain.
对于具有概率密度函数 f(x) 的连续随机变量 X，其累积分布函数 F(x) 定义为 F(x) = P(X ≤ x) = ∫-∞x f(t) dt。CDF 是一个非递减函数，随着 x 的增大从 0 上升到 1。实际操作中，你会从定义域的下限开始对 PDF 积分，一直积到 x。Edexcel 题目经常要求你针对不同的 x 范围写出 F(x) 的表达式，通常得到的是一个分段函数，在定义域下方的值为 0，在定义域上方的值为 1。
satisfies F(Q₃) = 0.75 and the lower quartile Q₁ satisfies F(Q₁) = 0.25. The difference Q₃ – range. These calculations are extremely common and require careful algebraic manipulation.
要求中位数 m，令 F(m) = 0.5 然后解出 m。上四分位数 Q₃ 满足 F(Q₃) = 0.75，下四分位数 Q₁ 满足 F(Q₁) = 0.25。差值 Q₃ – Q₁ 即为四分位距。这些计算极其常见，且需要细致的代数处理。
5. Finding Probabilities Directly from the PDF | 从PDF直接求概率
l [a, b] within the domain, probability is simply the area under the PDF: P(a < X < b) = ∫ab b) = P(a < X < b). You can also use the CDF to compute probabilities: P(a < X < b) = F(b) – ). In many exam questions, you will be required to find a probability such as P(X > c) = 1 – F(c).
对于定义域内的任意区间 [a, b]，概率就是 PDF 曲线下的面积：P(a < X < b) = ∫ab f(x) dx。由于连续随机变量在任意精确值上的概率为零，因此 P(X = a) = 0，且 P(a ≤ X ≤ b) = P(a < X < b)。你也可以利用 CDF 来计算概率：P(a < X < b) = F(b) – F(a)。在许多考题中，你会被要求求诸如 P(X > c) = 1 – F(c) 这样的概率。
| Probability expression | Using PDF | Using CDF |
| P(a ≤ X ≤ b) | ∫ab f(x) dx | F(b) – F(a) |
| P(X > c) | ∫c∞ f(x) dx | 1 – F(c) |
| P(X < d) | ∫-∞d f(x) dx | F(d) |
6. Mean and Expectation (E(X)) | 均值与期望
The mean or expected value of a continuous random variable X is given by E(X) = μ = ∫R ways confirm by integration. For a function of X, say g(X), the expectation is E(g(X)) = ∫R g(x) f(x) dx. This is the key formula needed to find E(X²), which later feeds into variance calculations.
连续随机变量 X 的均值或期望值由 E(X) = μ = ∫R x f(x) dx 给出，积分区间为整个定义域。如果 PDF 是对称的，均值常与对称轴重合，但你应该总是通过积分来确认。对于 X 的函数，比如 g(X)，其期望为 E(g(X)) = ∫R g(x) f(x) dx。这正是求 E(X²) 所需的关键公式，而 E(X²) 随后会用于方差的计算。
Always integrate carefully and double-check your limits. In Edexcel papers, expectation questions are often combined with finding constants, so a structured approach saves time: first determine the constant k, then compute E(X), and finally E(X²).
积分时务必仔细，并复核积分限。在Edexcel试卷中，期望问题常与求常数结合在一起，因此有条理的方法能节省时间：先确定常数 k，然后计算 E(X)，最后计算 E(X²)。
7. Variance and Standard Deviation | 方差与标准差
The variance of a continuous random variable X is defined as Var(X) = E(X²) – the variance, σ = √Var(X). Be prepared to give your final answer to three significant figures unless otherwise instructed. Also keep an eye out for questions that ask you to verify that Var(aX + b) = a² Var(X) for linear transformations of continuous variables.
连续随机变量 X 的方差定义为 Var(X) = E(X²) – [E(X)]²。首先通过积分计算 E(X) 和 E(X²)，然后代入公式。标准差就是方差的平方根，σ = √Var(X)。除非另有说明，你应该将最终答案保留三位有效数字。此外，要注意那些要求验证线性变换关系 Var(aX + b) = a² Var(X) 的题目。
Var(X) = ∫ x² f(x) dx – (∫ x f(x) dx)²
Pay special attention to algebraic mistakes when squaring E(X); many marks are lost here. Do the integration of x² f(x) first, then subtract the square of the mean.
在对 E(X) 进行平方时要特别注意代数错误，很多分数正是在这里丢掉了。先完成 x² f(x) 的积分，再减去均值的平方。
8. Mode and Median of a Continuous Distribution | 连续分布的众数与中位数
alue of x that maximises f(x) within the domain. To find it, you differentiate f(x), set f' ard. The median, as already mentioned, solves F(m) = 0.5. When the PDF consists of two pieces, you may need to decide which branch of the CDF to use by checking where the CDF reaches 0.5.
连续分布的众数是使得 f(x) 在定义域内取得最大值的 x 值。为求众数，你需要对 f(x) 求导，令 f'(x) = 0，然后用二阶导数检验或计算端点值。若 PDF 是单峰且可微的，这样做很直接。至于中位数，如前所述，满足 F(m) = 0.5。当 PDF 由两段组成时，你可能需要先判断 CDF 在哪一段达到 0.5，再选择相应的表达式。
Always state clearly that the mode is a maximum of the PDF, not the CDF. Many students confuse the two. In exam situations, if the PDF is given as a simple polynomial, the mode is often at a stationary point; however, if f(x) is monotonic, the mode will be at one of the interval endpoints.
务必明确指出众数是 PDF 的最大值点，而不是 CDF 的最大值点。很多学生会把两者搞混。在考试中，如果 PDF 被给定为一个简单的多项式，众数通常位于驻点处；然而，如果 f(x) 是单调的，那么众数将出现在区间的某个端点。
9. The Continuous Uniform Distribution | 连续均匀分布
random variable X is uniformly distributed over [a, b] if its PDF is constant: f(x) = 1/(b – a ≤ x ≤ b, and 0 otherwise. The mean is E(X) = (a + b)/2, and the variance is Var(X) = (b – a)²/12. The CDF is F(x) = (x – a)/(b – a) for a ≤ x ≤ b, and it provides a quick way to compute probabilities.
Edexcel S2 中经常出现的一个特例是连续均匀分布。若随机变量 X 在区间 [a, b] 上均匀分布，其 PDF 为常数：f(x) = 1/(b – a)，a ≤ x ≤ b，其他地方为 0。其均值为 E(X) = (a + b)/2，方差为 Var(X) = (b – a)²/12。其 CDF 为 F(x) = (x – a)/(b – a)（a ≤ x ≤ b），这为快速计算概率提供了便利。
Questions on the uniform distribution often ask you to find probabilities of intervals, the median, or percentiles using the CDF. For example, the median is exactly (a + b)/2, and the interquartile range is (b – a)/2. Memorising these standard results can save valuable minutes in an exam.
有关均匀分布的题目，常常要求你利用 CDF 求区间概率、中位数或百分位数。例如，中位数恰好是 (a + b)/2，四分位距为 (b – a)/2。熟记这些标准结果可以在考试中为你节省宝贵的答题时间。
10. Solving Exam-Style Problems Step by Step | 逐步攻克考试题型
Edexcel probability density function questions typically combine several of the above concepts into a single, multi-part problem. A classic structure is: (a) show that the constant k takes a given value, (b) find the cumulative distribution function, (c) determine the median or the interquartile range, (d) calculate P(0.5 < X < 1.5), (e) find E(X) and Var(X). Tackle each part sequentially, showing clear integration steps and algebraic simplification.
Edexcel 概率密度函数问题通常会将上述若干概念整合到一个多部分的题目中。经典结构是：(a) 证明常数 k 为给定值，(b) 求累积分布函数，(c) 确定中位数或四分位距，(d) 计算 P(0.5 < X < 1.5)，(e) 求 E(X) 和 Var(X)。依次解决每个部分，展示清晰的积分步骤和代数化简。
- Always sketch the PDF if it helps you visualise areas and probabilities. | 如果有助于直观理解面积和概率，一定要画出 PDF 的草图。
- Write down the CDF piecewise, clearly stating the value 0 for x below the domain and 1 for x above the domain. | 分段写出 CDF，明确标明 x 小于定义域时值为 0，大于定义域时值为 1。
- Use technology such as the calculator’s integration function to check your manual working, but always show your method. | 利用计算器的积分功能核对手算结果，但一定要展示解题过程。
- Pay attention to notation: use F(x) for CDF and f(x) for PDF. Never confuse the two. | 注意符号使用：CDF 用 F(x)，PDF 用 f(x)，切勿混淆。
When dealing with piecewise PDFs, split integrals at the points where the formula changes. If a PDF is defined by two different expressions on adjacent intervals, the total probability across both must sum to 1, and the CDF must be continuous (if the PDF is integrable). Exam mark schemes heavily penalise missing these continuity conditions.
处理分段 PDF 时，要在公式发生变化的点拆分积分。如果 PDF 在相邻区间上由两个不同的表达式定义，那么两个区间上的总概率之和必须等于 1，且 CDF 必须连续（如果 PDF 可积的话）。若忽略这些连续性条件，阅卷标准会严厉扣分。
11. Common Mistakes and How to Avoid Them | 常见错误与避免方法
he mode, or mistakenly differentiating F(x) and setting it to zero, believing that gives the mode. Remember: mode relates to f(x), not F(x). Also, when computing Var(X) = E(X²) – rly, causing an inaccurate final variance. Keep values in exact form until the final step.
许多学生求出常数后忘记检查 f(x) ≥ 0，这可能导致 PDF 无效。另一个常见错误是用 CDF 去求众数，或是错误地对 F(x) 求导并令其为零，以为那样就能得到众数。请记住：众数与 f(x) 有关，而非 F(x)。此外，在计算 Var(X) = E(X²) – [E(X)]² 时，学生常常过早地对 E(X) 进行舍入，导致最终的方差不准确。在最后阶段之前，应保持数值的精确形式。
n the definition you are using before substituting numbers. For instance, explicitly state "F(x) = ∫-∞x f(t) dt" and then substitute the appropriate lower limit. This reduces careless slip-ups and gives the examiner a clear path to follow, earning method marks even if you make an arithmetic mistake later.
要有条理：先写出你所使用的定义，再代入数字。例如，明确写出“F(x) = ∫-∞x f(t) dt”，然后代入正确的下限。这样做能减少粗心大意的错误，并为阅卷老师提供清晰的解题思路；即使后续出现算术错误，也能赢得过程分。
12. Connecting PDFs to Real-World Applications & Further Steps | PDF的实际应用与后续学习
Understanding probability density functions is not just about passing exams; it is the foundation for statistical inference, hypothesis testing, and more advanced topics like the normal distribution approximation and moment generating functions. The continuous uniform, normal, and exponential distributions all rely on PDFs, and mastering the core principles here will make those future topics significantly more approachable.
理解概率密度函数不仅仅是为了通过考试；它还是统计推断、假设检验以及正态分布逼近、矩母函数等更高级课题的基础。连续均匀分布、正态分布和指数分布都依赖于 PDF，掌握这里的核心原理将使你未来接触那些主题时轻松得多。
Practice past Edexcel questions consistently, varying the function types: linear, quadratic, square root, and trigonometric PDF fragments. The more you integrate, the quicker and more accurate you will become. Keep a formula compendium with standard results for E(X) and Var(X) of the uniform distribution, and don’t forget the power of symmetry to halve your work when a PDF is even about a central value.
持续练习Edexcel历年真题，变换函数类型：线性、二次、平方根以及三角形式的PDF片段。积分得越多，你的速度和准确性就越高。准备一本公式摘要，记录均匀分布的 E(X) 与 Var(X) 等标准结果，并且不要忘记利用对称性：当 PDF 关于某一中心值呈偶函数时，可以利用对称性让工作量减半。
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