Mastering Probability Density Functions for Edexcel A-Level Maths | 掌握概率密度函数：Edexcel 数学考点精讲

📚 Mastering Probability Density Functions for Edexcel A-Level Maths | 掌握概率密度函数：Edexcel 数学考点精讲

Probability density functions (PDFs) are a central topic in the Edexcel A-Level Mathematics statistics component. They allow us to model continuous random variables and calculate probabilities in a smooth, infinite sample space. This article provides a structured, exam-focused guide to understanding PDFs, their properties, and their applications.

概率密度函数（PDF）是 Edexcel A-Level 数学统计部分的核心主题。它使我们能够对连续型随机变量建模，并在光滑的无限样本空间中计算概率。本文提供一份结构化、紧扣考点的指南，帮助你理解 PDF 的定义、性质与应用。

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1. What Is a Continuous Random Variable? | 什么是连续随机变量？

A continuous random variable can take any value in an interval on the real number line. Examples include height, weight, time, and temperature. Unlike discrete variables, we cannot list all possible values, and the probability of any exact single value is zero.

连续随机变量可以取实数轴上某个区间内的任意值，例如身高、体重、时间和温度。与离散变量不同，我们无法列出所有可能取值，而且取到某一具体数值的概率为零。

The probability of a continuous random variable is described only over intervals, using a probability density function.

连续随机变量的概率只能通过概率密度函数在区间上来描述。

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2. Definition of the Probability Density Function (pdf) | 概率密度函数的定义

A probability density function f(x) is a function that describes the relative likelihood of a continuous random variable X taking a value near x. The total area under the curve f(x) over the whole range must equal 1.

概率密度函数 f(x) 刻画了连续随机变量 X 在 x 附近取值的相对可能性。曲线 f(x) 在全部范围内的总面积为 1。

∫ f(x) dx = 1 (over the whole range of X)

The domain of f(x) is the set of values for which f(x) > 0. Outside this domain, f(x) = 0.

f(x) 的定义域是使 f(x) > 0 的取值集合；在此范围之外，f(x) = 0。

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3. Conditions for a Valid Probability Density Function | 有效概率密度函数的条件

For a function f(x) to be a valid pdf on an interval [a, b], it must satisfy two key conditions:

要使函数 f(x) 成为区间 [a, b] 上的有效概率密度函数，必须满足两个关键条件：

• f(x) ≥ 0 for all x in the interval.

  在整个区间内 f(x) ≥ 0。

• The total area under the curve is exactly 1: ∫ₐᵇ f(x) dx = 1.

  曲线下的总面积为 1：∫ₐᵇ f(x) dx = 1。

These conditions are the first thing to check when finding an unknown constant.

在求解未知常数时，首先要检查这两个条件。

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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. It is the integral of the pdf from the lower bound up to x.

累积分布函数 F(x) 给出 X 小于或等于某个值的概率，即 pdf 从下限到 x 的积分。

F(x) = P(X ≤ x) = ∫ₐˣ f(t) dt

The CDF is useful because probabilities over intervals can be found as F(b) – F(a).

CDF 的用途在于，区间概率可以通过 F(b) – F(a) 求出。

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5. Finding Probabilities | 计算概率

For a continuous random variable, the probability that X lies between two values c and d is the area under the pdf between those values.

对于连续随机变量，X 落在 c 和 d 之间的概率是 pdf 在这两点之间的面积。

P(c ≤ X ≤ d) = ∫ f(x) dx, evaluated from c to d

Remember that for a continuous distribution, strict inequalities do not change the probability: P(c < X < d) = P(c ≤ X ≤ d).

请记住，对于连续分布，严格不等式不改变概率：P(c < X < d) = P(c ≤ X ≤ d)。

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6. Mean (Expected Value) and Variance | 均值（期望）与方差

The mean μ of a continuous random variable is given by the weighted integral of x with respect to the pdf.

连续随机变量的均值 μ 由 x 与 pdf 的加权积分给出。

μ = E(X) = ∫ x f(x) dx

The variance is found using E(X²) – [E(X)]².

方差通过 E(X²) – [E(X)]² 计算。

Var(X) = E(X²) – μ², where E(X²) = ∫ x² f(x) dx

These integrals are evaluated over the full range of X.

这些积分在 X 的全部范围内求值。

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7. Mode and Median | 众数和中位数

The mode is the value of x that maximises the pdf f(x). To find it, differentiate f(x) and set the derivative to zero, or inspect turning points within the domain.

众数是使 pdf f(x) 取得最大值的 x 值。求法是对 f(x) 求导，令导数为零，或检查定义域内的极值点。

The median m is the value that satisfies P(X ≤ m) = 0.5, which means the integral from the lower bound to m equals 0.5.

中位数 m 满足 P(X ≤ m) = 0.5，即下限到 m 的积分等于 0.5。

∫ₐᵐ f(x) dx = 0.5

Be careful: The mean, median and mode are not the same in a skewed distribution.

注意：在偏态分布中，均值、中位数与众数并不相同。

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8. Finding Unknown Constants with Integration | 用积分求解未知常数

Exam questions often give a pdf with an unknown constant k and ask you to find its value using the normalisation condition.

考试题中常常给出含未知常数 k 的 pdf，并要你用归一化条件求 k 的值。

Example: f(x) = kx² for 0 ≤ x ≤ 2, and 0 otherwise. To find k, set the integral over [0,2] equal to 1.

例：f(x) = kx²（0 ≤ x ≤ 2），否则为 0。求 k 时，令在 [0,2] 上的积分为 1。

∫₀² kx² dx = 1 → k [x³/3]₀² = 1 → k(8/3) = 1 → k = 3/8

Always state the domain before writing the final expression for f(x).

在写 f(x) 的最终表达式前，务必说明定义域。

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