📚 IB Mathematics: Distribution Functions – Definition and Calculation Methods | IB数学：分布函数的定义与计算方法
The concept of a distribution function, often called the cumulative distribution function (CDF), is central to probability and statistics. In IB Mathematics, both at Standard Level (SL) and Higher Level (HL), students are expected to understand its definition, properties, and how to use it to calculate probabilities for discrete and continuous random variables. This article provides a structured guide to mastering distribution functions, with clear examples and exam-focused tips.
分布函数（通常称为累积分布函数，CDF）是概率论与统计学的核心概念。在IB数学中，无论是标准水平（SL）还是高级水平（HL），学生都需要理解其定义、性质，并学会利用它来计算离散型和连续型随机变量的概率。本文将提供一份结构清晰的指南，帮助你掌握分布函数，并通过典型例题和考试技巧提升得分能力。
1. Random Variables and Probability Distributions | 随机变量与概率分布
A random variable is a variable whose possible values are outcomes of a random phenomenon. We usually denote it by a capital letter such as X. There are two main types: discrete random variables, which take distinct countable values, and continuous random variables, which take values in an interval. The probability distribution of a random variable describes how probabilities are assigned to its possible values.
随机变量是取值随随机现象结果而变化的变量，通常用大写字母如X表示。主要有两类：离散型随机变量取值是有限或可列个离散值；连续型随机变量则在一个区间内取值。随机变量的概率分布描述了概率如何分配给它的各个可能取值。
A probability distribution can be expressed in several forms: a table, a graph, a formula, or a distribution function. In IB Mathematics, the term "distribution function" almost always refers to the cumulative distribution function (CDF), denoted F(x).
概率分布可以通过多种形式表示：表格、图像、公式或分布函数。在IB数学中，“分布函数”一词几乎总是指累积分布函数，记为F(x)。
2. Definition of the Cumulative Distribution Function | 累积分布函数的定义
For a random variable X, the cumulative distribution function F(x) is defined as the probability that X takes a value less than or equal to x:
对于随机变量X，累积分布函数F(x)定义为X取值小于或等于x的概率：
F(x) = P(X ≤ x)
mming or integrating all probabilities from the lowest possible value up to and including x.
这个定义同时适用于离散型和连续型随机变量。符号P(X ≤ x)表示将自最小可能值到x（含x）的所有概率进行求和或积分。
+ 1/6 = 1/2. This simple idea is the foundation of all distribution function calculations.
例如，若X是掷一枚均匀六面骰子的点数，则F(3) = P(X ≤ 3) = P(X=1) + P(X=2) + P(X=3) = 1/6 + 1/6 + 1/6 = 1/2。这个简单的思想是所有分布函数计算的基础。
3. Properties of the Cumulative Distribution Function | 累积分布函数的性质
The cumulative distribution function has several key properties that are essential for solving IB problems. Recognising these properties helps you verify whether a given function is a valid CDF and use it correctly.
累积分布函数具有若干关键性质，这些性质对解决IB考试题目至关重要。理解这些性质有助于你验证给定函数是否为合法的CDF，并正确使用它。
F(x) is non-decreasing: if a < b then F(a) ≤ F(b).
F(x)是单调不减的：若 a < b，则 F(a) ≤ F(b)。
Limits: as x → -∞, F(x) → 0; as x → +∞, F(x) → 1.
极限性质：当 x → -∞ 时，F(x) → 0；当 x → +∞ 时，F(x) → 1。
F(x) is right-continuous, meaning that if you approach a value from the right, the function takes that same value.
F(x)是右连续的，即从右侧逼近某个值时，函数值等于该点的函数值。
For any real a and b with a < b, P(a < X ≤ b) = F(b) - F(a).
对任意实数a和b，若a < b，则 P(a < X ≤ b) = F(b) - F(a)。
These properties are universal, regardless of whether X is discrete or continuous. In IB questions, you may be asked to state these properties or use them to derive probabilities.
这些性质具有普遍性，无论X是离散型还是连续型都成立。在IB题目中，你可能会被要求叙述这些性质或利用它们推导概率。
4. Distribution Functions for Discrete Random Variables | 离散型随机变量的分布函数
or p(x). The cumulative distribution function is the running total of these probabilities:
对于离散型随机变量X，其概率由概率质量函数（PMF）给出，通常写作P(X = x)或p(x)。累积分布函数是这些概率的累积和：
F(x) = Σt ≤ x P(X = t)
where the sum includes all values t that are less than or equal to x and that X can take.
其中求和包含所有X可能的取值t，且满足 t ≤ x。
as x=2.5, F(x) is still 0.7, because no probability is assigned to values between 2 and 3.
例如，设X的PMF为：P(X=1)=0.2，P(X=2)=0.5，P(X=3)=0.3。则F(1)=0.2，F(2)=0.7，F(3)=1.0。对于2到3之间的任意x，如x=2.5，F(x)仍然等于0.7，因为在2和3之间没有分配概率。
A common IB task is to construct the CDF table from a given PMF, or vice versa. Remember that probabilities in a PMF are always non-negative and sum to 1.
IB考试中常见任务是根据给定的PMF构建CDF表格，或反过来做。记住PMF中的概率总是非负的，且总和为1。
5. Distribution Functions for Continuous Random Variables | 连续型随机变量的分布函数
. The cumulative distribution function is defined as the integral of the PDF from -∞ to x:
对于连续型随机变量X，X恰好取某个具体值的概率为零。取而代之，概率用概率密度函数（PDF）描述，记为f(x)。累积分布函数定义为PDF从 -∞ 到 x 的积分：
F(x) = ∫-∞x f(t) dt
Since probabilities over the entire sample space sum to 1, the total area under the PDF curve equals 1.
由于整个样本空间的总概率为1，PDF曲线下的总面积为1。
< 0, F(x) = x/2 for 0 ≤ x ≤ 2, and F(x) = 1 for x > 2. This piecewise function is the CDF.
例如，若一个连续型随机变量的PDF为 f(x) = 1/2（当0 ≤ x ≤ 2），其他情况为0，则 F(x) = 0（当x < 0），F(x) = x/2（当0 ≤ x ≤ 2），F(x) = 1（当x > 2）。这个分段函数就是CDF。
6. Relationship Between PDF and CDF | 概率密度函数与累积分布函数的关系
The relationship between the PDF and the CDF is fundamental in continuous probability. If the CDF is differentiable at x, then the PDF is its derivative:
PDF与CDF之间的关系是连续型概率论的基础。若CDF在x处可导，则PDF就是它的导数：
f(x) = F'(x) = dF/dx
Conversely, the CDF is the antiderivative of the PDF. In IB HL, you may be asked to find the PDF by differentiating a piecewise-defined CDF, or to find the CDF by integrating a PDF, remembering to include the constant of integration determined by boundary conditions.
反之，CDF是PDF的反导数。在IB高级水平中，你可能会被要求通过微分分段定义的CDF来求PDF，或通过积分PDF来求CDF，并利用边界条件确定积分常数。
Example: Given the CDF F(x) = 1 - e-3x for x ≥ 0, the PDF is f(x) = d/dx (1 - e-3x) = 3e-3x for x ≥ 0. This is an exponential distribution with rate λ = 3.
例如：已知CDF为 F(x) = 1 - e-3x（当x ≥ 0），则PDF为 f(x) = d/dx (1 - e-3x) = 3e-3x（当x ≥ 0）。这是一个速率为λ = 3的指数分布。
7. Calculating Probabilities Using the CDF | 利用累积分布函数计算概率
The most direct application of the CDF is computing probabilities over intervals. For a random variable X with CDF F(x):
CDF最直接的应用是计算区间概率。对于CDF为F(x)的随机变量X：
P(a < X ≤ b) = F(b) - F(a)
clusion or exclusion of endpoints changes the result because individual points have positive probability.
对于连续型X，严格不等式无所谓，所以 P(a < X < b) = P(a ≤ X ≤ b) = F(b) - F(a)。对于离散型X，端点是否包含会影响结果，因为单点具有正概率。
Discrete: P(a ≤ X ≤ b) = F(b) - F(a - 1) if X takes integer values.
离散型：若X取整数值，则 P(a ≤ X ≤ b) = F(b) - F(a - 1)。
Discrete: P(X > b) = 1 - F(b).
离散型：P(X > b) = 1 - F(b)。
Continuous: P(X ≤ a) = F(a), P(X > a) = 1 - F(a).
连续型：P(X ≤ a) = F(a)，P(X > a) = 1 - F(a)。
Always check whether the random variable is discrete or continuous before applying these formulas. IB examiners often include traps involving endpoints for discrete variables.
在应用这些公式前，务必判断随机变量是离散型还是连续型。IB考官经常针对离散型变量的端点设置陷阱。
8. Quantiles and the Inverse Distribution Function | 分位数与逆分布函数
and 1, the p-quantile is the smallest value x such that F(x) ≥ p. It is often written as F-1(p). In IB, you may need to find the median (p = 0.5) or other percentiles.
逆分布函数又称分位数函数，是另一个重要概念。对给定的概率p（0 < p < 1），p分位数是使得F(x) ≥ p的最小值x，通常记为F-1(p)。在IB中，你可能需要求中位数（p = 0.5）或其他百分位数。
For continuous random variables, if F is strictly increasing, then F-1 mple, if F(x) = x² for 0 ≤ x ≤ 1, then the median is found by solving x² = 0.5, giving x = √0.5 ≈ 0.707.
对于连续型随机变量，若F严格递增，则F-1(p)是满足F(x) = p的x值。例如，若F(x) = x²（0 ≤ x ≤ 1），则中位数通过解x² = 0.5得到，即x = √0.5 ≈ 0.707。
used in finding the lower quartile, median, and upper quartile of a discrete distribution.
对于离散型随机变量，逆CDF的意义略有不同，因为F是阶梯函数。p分位数是满足F(x) ≥ p的最小x。这常用于求离散分布的下四分位数、中位数和上四分位数。
9. Expected Value and Variance Using the CDF | 利用累积分布函数求期望与方差
Although expected value and variance are usually calculated from the PMF or PDF, it is also possible to express them in terms of the CDF. For a non-negative continuous random variable, the expected value can be written as:
尽管期望和方差通常由PMF或PDF计算，但也存在用CDF表达的公式。对于非负的连续型随机变量，期望可写为：
E[X] = ∫0∞ (1 - F(x)) dx
For a general continuous random variable with support on the whole real line, the mean can be expressed using integrals involving F(x) and 1 - F(x). However, in IB, this formula appears mainly at HL or in enrichment contexts; standard calculations use the PMF or PDF directly.
对于整个实数轴上有定义的连续随机变量，其均值可用含F(x)和1 - F(x)的积分表示。但在IB中，这个公式主要出现在高级水平或拓展内容中；标准计算仍直接使用PMF或PDF。
CDF and asked to find the PDF, then compute E[X] and Var(X). For example, if F(x) = 1 - e-2x for x ≥ 0, then f(x) = 2e-2x, which is an exponential distribution with mean 1/2 and variance 1/4. You should recognise standard distributions to save time.
更常见的情况是给出CDF，让你先求PDF，再计算E[X]和Var(X)。例如，若F(x) = 1 - e-2x（x ≥ 0），则f(x) = 2e-2x，这是指数分布，均值为1/2，方差为1/4。你应该熟悉常见分布以节省时间。
10. Worked Example: Discrete Distribution | 典例解析：离散型分布
Consider a random variable X with the following probability mass function:
考虑一个随机变量X，其概率质量函数如下：
| x | 0 | 1 | 2 | 3 |
| P(X = x) | 0.1 | 0.2 | 0.4 | 0.3 |
Find the CDF and calculate P(1 < X ≤ 3).
求CDF，并计算P(1 < X ≤ 3)。
3) - F(1) = 1.0 - 0.3 = 0.7. Alternatively, this equals P(X=2) + P(X=3) = 0.4 + 0.3 = 0.7.
解答：F(0) = 0.1，F(1) = 0.1 + 0.2 = 0.3，F(2) = 0.3 + 0.4 = 0.7，F(3) = 0.7 + 0.3 = 1.0。因此，P(1 < X ≤ 3) = F(3) - F(1) = 1.0 - 0.3 = 0.7。另一种算法：P(X=2) + P(X=3) = 0.4 + 0.3 = 0.7。
equality is strict on the left. This distinction is crucial for discrete random variables.
注意P(1 < X ≤ 3)不包含P(X=1)，因为左边是严格不等。这种区分对离散型随机变量至关重要。
11. Worked Example: Continuous Distribution | 典例解析：连续型分布
Let X be a continuous random variable with probability density function:
设X是一个连续型随机变量，其概率密度函数为：
f(x) = kx² for 0 ≤ x ≤ 2, and 0 otherwise
Find the value of k, the CDF F(x), and P(0.5 < X < 1.5).
求k的值、CDF F(x)以及P(0.5 < X < 1.5)。
First, the total area under the PDF must be 1:
首先，PDF曲线下的总面积必须等于1：
∫02 kx² dx = k [x³/3]02 = 8k/3 = 1
Therefore k = 3/8. The CDF for 0 ≤ x ≤ 2 is:
因此 k = 3/8。当0 ≤ x ≤ 2时，CDF为：
F(x) = ∫0x (3/8)t² dt = x³/8
Thus F(x) = 0 for x < 0, F(x) = x³/8 for 0 ≤ x ≤ 2, and F(x) = 1 for x > 2. Then:
因此 F(x) = 0（当x < 0），F(x) = x³/8（当0 ≤ x ≤ 2），F(x) = 1（当x > 2）。进而：
< X < 1.5) = F(1.5) - F(0.5) = (1.5)³/8 - (0.5)³/8 = 3.375/8 - 0.125/8 = 3.25/8 = 0.40625
This method avoids integrating the PDF twice, saving valuable exam time.
这种方法避免了两次积分PDF，从而节省宝贵的考试时间。
12. Common Pitfalls and Exam Tips | 常见误区与考试技巧
IB students often lose marks on distribution function questions because of small but avoidable errors. The following list highlights the most common pitfalls and how to avoid them.
IB学生在分布函数题目中常常因为一些细小但可以避免的错误而丢分。以下列出了最常见的误区和应对方法。
nd CDF. The PMF gives P(X = x), while the CDF gives P(X ≤ x). Read the question carefully.
混淆PMF和CDF。PMF给出P(X = x)，而CDF给出P(X ≤ x)。仔细审题。
Forgetting piecewise definitions. A CDF is often defined piecewise. Always state the domain of each piece, including x < lower bound and x > upper bound.
忘记分段定义。CDF通常是分段函数。务必写出每一段的定义域，包括x小于下界和x大于上界的情况。
rict inequalities incorrectly. Remember F(b) - F(a) equals P(a < X ≤ b), not P(a ≤ X < b).
在离散型变量中使用严格不等式出错。记住F(b) - F(a)等于P(a < X ≤ b)，而不等于P(a ≤ X < b)。
When finding the PDF from a CDF, forgetting that the derivative of a constant piece is 0, and that boundary points are included with zero probability for continuous variables.
由CDF求PDF时，忘记常数段的导数为0；对于连续型变量，端点概率为0。
Using the inverse CDF incorrectly. For a discrete distribution, F-1(p) is not always the solution to F(x) = p, because F is a step function.
错误使用逆CDF。对于离散分布，F-1(p)并不总是方程F(x) = p的解，因为F是阶梯函数。
In exams, always sketch a quick graph of the CDF if possible. This helps you verify monotonicity and check whether the final probability makes sense. For continuous variables, the graph of the CDF should be smooth (except possibly at a finite number of points).
在考试中，如果可能，快速画出CDF的草图。这有助于验证单调性，并检查最终概率是否合理。对于连续型变量，CDF图像应当是光滑的（除了可能在有限个点处不光滑）。
1. If your computed CDF does not satisfy these limits, you have made a calculation error.
最后，务必检查F(-∞) = 0和F(+∞) = 1。如果你计算出的CDF不满足这些极限，说明计算有误。
Mastering distribution functions is not just about memorising formulas; it is about understanding how probability accumulates across the possible values of a random variable. By practising with both discrete and continuous examples, and by paying attention to inequalities and domain restrictions, you will build the skills needed to solve IB problems confidently and accurately.
掌握分布函数不仅仅在于记忆公式，更在于理解概率如何随着随机变量的取值而累积。通过练习离散型和连续型例题，并关注不等式与定义域限制，你将具备自信而准确地解决IB相关问题的能力。
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