The cumulative distribution function | 累积分布函数

📚 The cumulative distribution function | 累积分布函数

In A-Level probability and statistics, the cumulative distribution function (CDF) provides a complete description of the probability structure of a random variable. Whether you are working with discrete or continuous data, the CDF tells you the probability that the random variable takes a value less than or equal to a given point. Understanding its definition, properties and applications is essential for tackling exam questions on distributions, probabilities, percentiles, and transformations involving random variables.

在 A-Level 概率与统计中，累积分布函数 (CDF) 完整地描述了一个随机变量的概率结构。无论你处理的是离散数据还是连续数据，CDF 都能告诉你该随机变量取值小于或等于某一点的概率。理解它的定义、性质和应用，对于解决关于分布、概率、百分位数以及涉及随机变量变换的考试题目至关重要。

1. What is the Cumulative Distribution Function? | 什么是累积分布函数？

. Formally, F(x) = P(X ≤ x), where x is any real number. For a discrete random variable, this is obtained by summing the probability mass function up to x; for a continuous random variable, it is the integral of the probability density function from the lower bound up to x. The CDF always exists, regardless of the type of random variable, and it is one of the most versatile tools in probability theory.

随机变量 X 的累积分布函数记为 F(x)，它给出了 X 小于或等于某个特定值 x 的概率。正式定义为 F(x) = P(X ≤ x)，其中 x 是任意实数。对于离散随机变量，这通过对概率质量函数求和直到 x 得到；对于连续随机变量，则是概率密度函数从下界到 x 的积分。无论随机变量是什么类型，CDF 始终存在，是概率论中最通用的工具之一。

F(x) = P(X ≤ x)

F(x) = P(X ≤ x)

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2. Basic Properties of the CDF | 累积分布函数的基本性质

Every cumulative distribution function has a set of characteristic features. It is a non-decreasing function, meaning that as x increases, F(x) never decreases. Its limits are such that as x approaches negative infinity, F(x) tends to 0, and as x approaches positive infinity, F(x) tends to 1. The CDF is also right-continuous; for any value c, the limit from the right equals F(c). In the continuous case the function is actually continuous everywhere. These properties ensure that the CDF properly models the accumulation of probability.

每一个累积分布函数都有一组特征。它是一个非递减函数，意味着随着 x 增加，F(x) 从不减少。它的极限是：当 x 趋近负无穷时，F(x) 趋于 0；当 x 趋近正无穷时，F(x) 趋于 1。CDF 还是右连续的；对任何 c，右极限等于 F(c)。在连续情形下，该函数实际上是处处连续的。这些性质确保了 CDF 正确地模拟了概率的累积过程。

English: Key limits: lim_x→−∞ F(x) = 0, lim_x→∞ F(x) = 1.

中文：关键极限：lim_x→-∞ F(x) = 0, lim_x→∞ F(x) = 1。

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3. CDF for Discrete Random Variables | 离散随机变量的累积分布函数

ained by summing probabilities for all outcomes less than or equal to x. That is, F(x) = ∑_t≤x p(t). The CDF of a discrete variable is a step function that jumps at each possible value of X by an amount equal to the probability of that value. Between jumps, the function remains constant.

对于具有概率质量函数 p(x) = P(X = x) 的离散随机变量 X，累积分布函数通过将所有小于等于 x 的结果的概率相加得到，即 F(x) = ∑_t≤x p(t)。离散变量的 CDF 是一个阶梯函数，在 X 的每一个可能取值处跳跃，跳跃的高度等于该值的概率。在两次跳跃之间，函数保持不变。

Example: 3 with probabilities 0.2, 0.5, 0.3, then F(1.5) = P(X≤1.5) = 0.2, F(2) = 0.2+0.5 = 0.7.

示例：若 X 取值 1,2,3，概率分别为 0.2,0.5,0.3，则 F(1.5) = P(X≤1.5) = 0.2，F(2) = 0.2+0.5 = 0.7。

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4. CDF for Continuous Random Variables | 连续随机变量的累积分布函数

e integral of the density over the interval from the minimum possible value to x: F(x) = ∫_−∞^x creasing curve with F(x) = 0 for x below the support and F(x) = 1 for x above the support.

当 X 是具有概率密度函数 f(x) 的连续随机变量时，累积分布函数定义为从最小可能值到 x 区间上密度的积分：F(x) = ∫_-∞^x f(t) dt。因为密度 f(t) 非负且在其定义域上积分为 1，所以 F(x) 是一条光滑、非递减的曲线，在支撑集下方 F(x) = 0，在支撑集上方 F(x) = 1。

F(x) = ∫_−∞^x f(t) dt

F(x) = ∫_-∞^x f(t) dt

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5. Deriving the CDF from the PDF | 从概率密度函数推导累积分布函数

defined in separate intervals. For each interval, you integrate the appropriate expression and ensure that the constants of integration are chosen so that F(x) is continuous and satisfies F(lower bound) = 0. The final step is to define F(x) = 1 for x beyond the upper limit of the support.

当只给出 PDF 时，需要逐段积分求得 CDF。许多考试题目会给出定义在不同区间上的概率密度函数。对每一个区间，你分别积分相应的表达式，并选择积分常数使得 F(x) 连续且满足 F(下界) = 0。最后一步是对于支撑集上限以外的 x 定义 F(x) = 1。

Worked snippet: f(x) = kx for 0≤x≤2, first find k using total area 1, then for 0≤x≤2, F(x) = ∫₀^x kt dt = (k/2) x². Below 0, F(x)=0; above 2, F(x)=1.

示例片段：若 f(x) = kx 当 0≤x≤2，先利用总面积 1 求出 k，然后对 0≤x≤2，有 F(x) = ∫₀^x kt dt = (k/2) x²。小于 0 时 F(x)=0；大于 2 时 F(x)=1。

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6. Computing Probabilities with the CDF | 使用累积分布函数计算概率

b) = F(b) − F(a). This formula works for both discrete and continuous variables, provided we account for the inclusive or exclusive boundaries correctly. For continuous variables, the distinction between < and ≤ does not matter because individual points have zero probability.

CDF 最实际的用途之一是求随机变量落在某个区间内的概率。对任意 a < b，有 P(a < X ≤ b) = F(b) – F(a)。该公式适用于离散和连续变量，只要正确处理包含或不包含的边界即可。对于连续变量，< 和 ≤ 之间的区别无关紧要，因为单个点的概率为零。

Example: − F(3), P(1 < X ≤ 4) = F(4) − F(1). These shortcuts save time in exams.

示例：P(X > 3) = 1 – F(3)，P(1 < X ≤ 4) = F(4) – F(1)。这些快捷方法在考试中能节省时间。

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7. The Median, Quartiles and Percentiles | 中位数、四分位数和百分位数

3 satisfies F(Q3) = 0.75. In general, the 100p-th percentile is the solution to F(x) = p, where 0<p<1. For discrete variables, you may need to use linear interpolation if the exact probability is not attained.

对于连续随机变量，中位数 m 是满足 F(m) = 0.5 的值。下四分位数 Q1 满足 F(Q1) = 0.25，上四分位数 Q3 满足 F(Q3) = 0.75。一般来说，第 100p 百分位数是方程 F(x) = p 的解，其中 0

rrect interval. The interquartile range IQR = Q3 − Q1 is a common measure of spread.

当 CDF 以分段函数形式给出时，令相应的分支等于 0.5 并求解，并验证解落在正确的区间内。四分位距 IQR = Q3 – Q1 是一种常用的离散程度度量。

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8. CDF of the Continuous Uniform Distribution | 连续均匀分布的累积分布函数

ution function grows linearly from 0 to 1 on the interval: F(x) = (x−a)/(b−a) for a≤x≤b, with F(x)=0 for x<a, and F(x)=1 for x>b. This simple form makes it an excellent example for practising CDF manipulation and finding medians, where m = (a+b)/2.

区间 [a,b] 上的连续均匀分布具有常数概率密度 f(x) = 1/(b–a)，其中 a≤x≤b。它的累积分布函数在该区间上从 0 线性增长到 1：当 a≤x≤b 时，F(x) = (x–a)/(b–a)；当 xb 时 F(x)=1。这种简单的形式使它成为练习 CDF 操作和求中位数（其中 m = (a+b)/2）的绝佳范例。

条件	F(x)
x < a	0
a ≤ x ≤ b	(x − a)/(b − a)
x > b	1

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9. CDF of the Exponential Distribution | 指数分布的累积分布函数

nential distribution with rate parameter λ (λ>0) has PDF f(x) = λe^−λx for x≥0. Its CDF is F(x) = 1 − e^−λx is function captures the memoryless property of the exponential distribution. It is frequently tested in Edexcel S2 and S3, especially in contexts involving waiting times or lifetimes.

参数为 λ（λ>0）的指数分布对于 x≥0 有 PDF f(x) = λe^–λx。它的 CDF 在 x≥0 时为 F(x) = 1 – e^–λx，而 x<0 时 F(x)=0。这个函数体现了指数分布的无记忆性。在 Edexcel S2 和 S3 的考题中经常出现，尤其是在涉及等待时间或寿命的情境中。

F(x) = 1 − e^−λx for x ≥ 0

F(x) = 1 – e^–λx 当 x ≥ 0

The median m is found by solving 1 − e^−λm s result is useful for comparing central tendency under different rates and for transforming data.

中位数 m 通过解 1 – e^–λm = 0.5 得到，m = (ln 2)/λ。该结果对于比较不同速率下的集中趋势以及数据变换非常有用。

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10. Worked Examples and Exam Tips | 例题与考试技巧

Let' erwise. First, verify the total area equals 1. Then integrate to find the CDF: for 1≤x≤3, F(x) = ∫₁^x (1/8)(1)³ = 7/8. Also find the median by solving (1/8)(m&minus;1)³ = 0.5, giving m = 1 + 4^1/3.

我们来解决一道典型的 Edexcel 风格题目。假设连续随机变量 X 具有 PDF f(x) = (3/8)(x–1)²，当 11^x (3/8)(t–1)² dt = (1/8)(x–1)³。小于 1 时 F(x)=0；大于 3 时 F(x)=1。接着可计算 P(X>2) = 1 – F(2) = 1 – (1/8)(1)³ = 7/8。再求中位数，解 (1/8)(m–1)³ = 0.5，得 m = 1 + 4^1/3。

Common pitfalls include forgetting to add the constant of integration, using the wrong branch of a piecewise CDF, or misapplying the probability formula for inclusive boundaries. Always check that your CDF is non-decreasing, starts at 0 and ends at 1, and is right-continuous. When sketching, label key points and jumps clearly.

常见错误包括忘记加上积分常数、使用分段 CDF 的错误分支，或对包含边界概率公式的错误应用。务必检查你的 CDF 是否非递减、从 0 开始到 1 结束，且右连续。在绘图时，清楚标出关键点和跳跃。

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