📚 Probability Density Functions | 概率密度函数
In A-Level Mathematics, moving from discrete random variables to continuous ones requires a shift in thinking. Instead of a probability mass function that lists individual outcomes, we use a probability density function (PDF) to describe how probabilities are distributed over an interval. The PDF is a fundamental tool in Edexcel Statistics, helping us model everything from waiting times to measurement errors.
在 A-Level 数学中，从离散随机变量过渡到连续随机变量需要转变思维方式。我们不再使用列举单独结果的概率质量函数，而是用概率密度函数 (PDF) 来描述概率在一个区间内的分布情况。PDF 是爱德思统计学的核心工具，帮助我们建立从等待时间到测量误差等各种模型。
1. Introduction to Continuous Random Variables | 连续随机变量简介
A continuous random variable can take any value within a given range. Unlike a discrete variable, the probability of it taking an exact single value is zero. Instead, we define probabilities over intervals. For example, the height of a randomly chosen adult or the lifetime of a light bulb are continuous quantities.
连续随机变量可以在给定范围内取任意值。与离散变量不同，它取某个精确单值的概率为零。因此，我们定义区间上的概率。例如，随机选取的成年人身高或灯泡的寿命，都是连续量。
The probability distribution of a continuous random variable X is described by a probability density function, denoted f(x). It must be integrated to find probabilities, rather than summed. This is why integration skills from Pure Mathematics become vitally important in Statistics units.
连续随机变量 X 的概率分布由概率密度函数（记作 f(x)）来描述。必须通过积分而不是求和来求概率。正因如此，纯数学中的积分技巧在统计单元中变得至关重要。
2. Definition of a Probability Density Function | 概率密度函数的定义
The probability that X lies between two values c and d is given by the area under the curve f(x) from c to d: P(c < X < d) = ∫cd f(x) dx.
概率密度函数 (PDF) 是定义在连续样本空间（通常是区间 [a, b] 或 (−∞, ∞)）上的非负函数 f(x)。X 落在两个值 c 与 d 之间的概率由曲线 f(x) 从 c 到 d 的下方面积给出：P(c < X < d) = ∫cd f(x) dx。
d) = P(c < X < d). This is a key conceptual difference from discrete distributions.
由于取精确值的概率为零，开区间和闭区间得到相同的结果。因此 P(c ≤ X ≤ d) = P(c < X < d)。这是与离散分布一个关键的概念差异。
3. Conditions for a Valid PDF | 有效PDF的条件
≥ 0 for all x in its domain, and (ii) the total area under the curve equals one, that is ∫all x f(x) dx = 1. These ensure probabilities are never negative and sum to unity.
要使 f(x) 成为一个有效的概率密度函数，必须满足两个条件：(i) 在其定义域内对所有 x 均满足 f(x) ≥ 0，(ii) 曲线下的总面积等于 1，即 ∫全体 x f(x) dx = 1。这确保概率永不为负且总和为一。
In exam questions, you are often given a function with an unknown constant k. You are required to set the integral of f(x) over the valid domain equal to 1, solve for k, and then check that f(x) remains non-negative. Never forget to verify the sign of f(x) after finding k.
在考题中，经常会给出一个带有未知常数 k 的函数。你需要令 f(x) 在有效区域上的积分等于 1，解出 k，然后检验 f(x) 保持非负。求出 k 后永远不要忘记检验 f(x) 的符号。
4. Finding Probabilities Using PDFs | 使用PDF求概率
To find P(a < X < b), evaluate the definite integral of f(x) between a and b. If the function is piecewise, ensure you only integrate over the region where f(x) is defined. Always draw a quick sketch of the PDF to visualise the required area; this reduces integration errors.
要求 P(a < X < b)，计算 f(x) 在 a 和 b 之间的定积分。如果函数是分段定义的，确保你只在 f(x) 有定义的部分积分。始终快速绘制 PDF 草图来直观显示所需面积；这能减少积分错误。
Because the total area is 1, using complements is efficient. For an upper tail, compute ∫bupper limit f(x) dx, where the upper limit is the maximum bound of the defined domain.
有时需要求区间外的概率，例如 P(X > b) = 1 − P(X < b)。由于总面积为 1，使用补集非常高效。对于上尾概率，计算 ∫b上限 f(x) dx，其中上限是定义域的最大界。
5. The Cumulative Distribution Function (CDF) | 累积分布函数(CDF)
), gives the probability that X takes a value less than or equal to x: F(x) = P(X ≤ x) = ∫−∞x ) is a continuous, non-decreasing function with F(lower bound) = 0 and F(upper bound) = 1.
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。对于连续变量，F(x) 是一个连续的非递减函数，满足 F(下界) = 0 和 F(上界) = 1。
If you are given a CDF, you can find probabilities directly: P(a < X < b) = F(b) − F(a).
CDF 在求中位数和四分位数时特别有用，也便于无需每次对 PDF 重新积分即可计算概率。如果给出了 CDF，可以直接求概率：P(a < X < b) = F(b) − F(a)。
6. Relationship Between PDF and CDF | PDF与CDF的关系
The PDF is the derivative of the CDF with respect to x: f(x) = F' on. This inverse relationship is often tested by asking you to convert one into the other.
PDF 是 CDF 对 x 的导数：f(x) = F'(x)，在导数存在的点成立。反过来，CDF 是 PDF 的积分，通常从分布的最小界限开始积分。这种互逆关系经常作为考点，要求你从一种形式转换到另一种。
Be careful with piecewise CDFs: differentiate each piece separately, and ensure the pieces of f(x) join naturally. Exam questions may present a CDF and ask you to confirm it is valid by showing monotonicity and endpoint conditions, then derive the PDF.
处理分段 CDF 时要小心：对每一段分别求导，并确保 f(x) 的各段自然衔接。考题可能会给出 CDF，让你通过证明单调性和端点条件来确认其有效性，然后推导出 PDF。
7. Median and Quartiles | 中位数和四分位数
The median m of a continuous distribution satisfies F(m) = 0.5, or equivalently ∫minm ower quartile Q1 satisfies F(Q1) = 0.25, and the upper quartile Q3 satisfies F(Q3) = 0.75.
连续分布的中位数 m 满足 F(m) = 0.5，也就是 ∫下限m f(x) dx = 0.5。它将分布分为面积相等的两部分。类似地，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 find a median from a PDF directly, set up the definite integral from the lower bound to an unknown m, equate it to 0.5, and solve for m. Always check that your solution lies inside the defined domain. If the PDF is symmetric, the median equals the point of symmetry.
直接从 PDF 求中位数时，建立从下界到未知数 m 的定积分，令其等于 0.5，再解出 m。务必检验解落在定义域内。如果 PDF 是对称的，中位数就等于对称点。
8. Mode of a Continuous Distribution | 连续分布的众数
ble is the value of x where f(x) attains its maximum within the defined domain. Unlike the mean and median, the mode is found by differentiation: set f' or x, then check end points and the second derivative or sign change to confirm a maximum.
连续随机变量的众数是 f(x) 在定义域内取得最大值的 x 值。与均值和中位数不同，众数通过微分求得：令 f'(x) = 0 解出 x，然后检查端点以及二阶导数或符号变化来确认最大值点。
Sometimes the PDF is strictly decreasing or increasing over its domain; then the mode occurs at a boundary. For a uniform distribution, every point in the interval is a mode because the density is constant. It is important to state clearly whether a distribution is unimodal or has multiple modes.
有时 PDF 在其定义域内严格递减或递增，那么众数就出现在边界上。对于均匀分布，区间内每一点都是众数，因为密度恒定。必须清楚地说明分布是单峰的还是具有多个众数。
9. Expectation and Variance | 期望和方差
)] = ∫ g(x) f(x) dx. The variance is Var(X) = E(X²) − [E(X)]², where E(X²) = ∫ x² f(x) dx.
连续随机变量 X 的期望值（均值）为在全体域上的 E(X) = ∫ x f(x) dx。对于函数 g(X)，期望为 E[g(X)] = ∫ g(x) f(x) dx。方差为 Var(X) = E(X²) − [E(X)]²，其中 E(X²) = ∫ x² f(x) dx。
Var(X) = ∫ x² f(x) dx - ( ∫ x f(x) dx )²
he variance formula. Standard deviation is simply the positive square root of variance, SD(X) = √Var(X).
这些公式是离散情况的直接类比，求和使用积分代替。永远先计算 E(X) 和 E(X²)，然后运用方差公式。标准差就是方差的正平方根，SD(X) = √Var(X)。
10. Linear Transformations of Continuous Random Variables | 连续随机变量的线性变换
r(X). The PDF of Y can be derived using either the CDF method or the change of variable method, but for linear transformations the mean and variance rules suffice for most Edexcel questions.
如果 Y = aX + b，其中 a 和 b 为常数，那么 E(Y) = aE(X) + b，Var(Y) = a²Var(X)。Y 的 PDF 可用 CDF 法或变量变换法推导，但对于线性变换，大多数爱德思题目只需使用均值和方差的规则即可。
e of typical A-Level questions. However, the expectation and variance rules for sums still hold: E(X+Y) = E(X)+E(Y) and Var(X+Y) = Var(X)+Var(Y) if X and Y are independent.
当两个独立的连续变量相加，例如 Z = X + Y，和的 PDF 是各自 PDF 的卷积，这通常超出了 A-Level 题目的范围。但和的期望与方差规则依然成立：若 X 与 Y 独立，则 E(X+Y) = E(X)+E(Y)，Var(X+Y) = Var(X)+Var(Y)。
11. Working with Piecewise PDFs | 分段PDF的处理
ate each piece over its subinterval and equate the sum to 1. Be methodical: write the integrals separately, add them, and solve.
许多考题中的 PDF 是分段定义的，例如 f(x) = kx 当 0 ≤ x < 2，f(x) = k(4-x) 当 2 ≤ x ≤ 4。为求 k，对每个分段在其子区间上积分，令总和为 1。要有条理：分别写出积分，相加，再求解。
When working with piecewise CDFs, you obtain F(x) by integrating the PDF from the minimum bound up to x, which results in different expressions on different intervals. Ensure F(x) is continuous at the join points. Discontinuity in F(x) would violate the definition of a continuous CDF.
处理分段 CDF 时，通过从最小界限到 x 积分 PDF 得到 F(x)，这会在不同区间上产生不同的表达式。确保 F(x) 在连接点处连续。CDF 的不连续性会违反连续型 CDF 的定义。
12. Common Exam Techniques and Pitfalls | 常见考试技巧与易错点
Always write down the domain for f(x) before integrating. A common mistake is to integrate beyond the defined region and obtain an area greater than 1, or to forget that the PDF is zero outside its domain. Use clear notation and show your substitution limits carefully.
在积分前务必写下 f(x) 的定义域。一个常见错误是积分超出定义区域而得到面积大于 1，或忘记 PDF 在定义域外为零。使用清晰的符号，并仔细标明换元积分的上下限。
s must be discarded. For variance, many candidates lose marks by forgetting to square the mean when applying Var(X) = E(X²) − μ².
当题目要求求中位数时，设 F(m)=0.5 并求解，但如果 F(x) 是分段函数，要选择正确的分支。也要检查负值解是否必须舍去。对于方差，许多考生因应用 Var(X) = E(X²) − μ² 时忘记对均值平方而丢分。
Finally, relate your answers back to context: interpret the probability or the expected value in the wording of the problem. This shows understanding and can earn communication marks. Practise mixing PDF and CDF manipulations quickly and accurately under timed conditions.
最后，将你的答案与题目背景联系起来：用问题的措辞解释概率或期望值。这体现出理解力，还能获得表达分。在限时条件下，练习快速而准确地混合运用 PDF 和 CDF 的变换。
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