📚 Continuous Random Variables and Distributions | 连续随机变量及其分布
Continuous random variables form a cornerstone of probability theory, extending the idea of randomness beyond countable outcomes to measurements on a continuous scale. Unlike discrete variables that jump between isolated values, a continuous random variable can take any value within an interval, such as time, length, or temperature. To describe their behaviour, we introduce probability density functions, cumulative distribution functions, and key characteristics such as expectation and variance. This article covers the essential concepts, properties, and common continuous distributions – including the uniform and normal distributions – that appear throughout the IB Mathematics curriculum.
连续随机变量是概率论的重要基石，它将随机性的概念从可数结果拓展到了连续尺度上的度量。与在孤立值之间跳跃的离散变量不同，连续随机变量可以在某个区间内取任意值，例如时间、长度或温度。为了描述它们的行为，我们引入了概率密度函数、累积分布函数以及期望和方差等关键特征。本文涵盖了IB数学课程中贯穿始终的基本概念、性质以及常见的连续分布——包括均匀分布和正态分布。
1. What is a Continuous Random Variable? | 什么是连续随机变量？
A continuous random variable is a variable whose set of possible outcomes is uncountable and typically forms an interval on the real number line. For example, the exact height of a randomly selected student or the waiting time at a bus stop are continuous quantities. We cannot list all possible values nor assign a non-zero probability to any single point; instead, probabilities are assigned to intervals.
连续随机变量是指可能结果的集合不可数、通常在实数轴上形成一个区间的变量。例如，随机选取一名学生的精确身高，或在公交车站的等候时间，都是连续的量。我们无法列出所有可能的值，也无法给任何单点赋予非零概率；相反，概率是赋予给区间的。
2. Probability Density Function (PDF) | 概率密度函数（PDF）
The probability density function, denoted f(x), describes the relative likelihood of a continuous random variable taking on a particular value. The probability that the variable X lies in the interval [a, b] is given by the area under f(x) from a to b: P(a ≤ X ≤ b) = ∫ab f(x) dx. Importantly, f(x) itself is not a probability; it can be greater than 1, but the total area under the curve must equal 1.
概率密度函数，记作f(x)，描述了连续随机变量取某个特定值的相对可能性。变量X落在区间 [a, b] 内的概率由 f(x) 从 a 到 b 的曲线下面积给出：P(a ≤ X ≤ b) = ∫ab f(x) dx。重要的是，f(x) 本身不是概率；它可以大于 1，但曲线下的总面积必须等于 1。
3. Properties of a Valid PDF | 有效概率密度函数的性质
omain; and (2) the total area under the curve over the entire range of X equals 1, i.e., ∫−∞∞ gle point c, so it does not matter whether endpoints are included in probability calculations.
一个函数 f(x) 要成为概率密度函数，必须满足两个条件：(1) 在其定义域内处处有 f(x) ≥ 0；(2) 在整个 X 取值范围内曲线下的总面积等于 1，即 ∫−∞∞ f(x) dx = 1。此外，对于连续变量，任何单点 c 的概率 P(X = c) = 0，因此在计算概率时是否包含端点并不影响结果。
4. Cumulative Distribution Function (CDF) | 累积分布函数（CDF）
The cumulative distribution function, F(x) lity that the random variable X takes a value less than or equal to x: F(x) = P(X ≤ x) = ∫−∞x cularly useful for computing probabilities over intervals without performing integration each time.
累积分布函数 F(x) 给出了随机变量 X 取值小于或等于 x 的概率：F(x) = P(X ≤ x) = ∫−∞x f(t) dt。F(x) 是一个非递减函数，从 0（当 x → −∞ 时）递增至 1（当 x → ∞ 时）。CDF 特别适合用于计算区间概率，而无需每次都进行积分运算。
5. Relationship Between PDF and CDF | PDF与CDF的关系
ion can be obtained from the cumulative distribution function by differentiation: f(x) = F' derivative of the CDF recovers the density, and integrating the density produces the CDF.
概率密度函数可以通过对累积分布函数求导得到：f(x) = F'(x)，其中 F 是可微的。反过来，CDF 是 PDF 的积分。这种互逆关系是基本的：CDF 的导数恢复出密度函数，而积分密度函数则产生 CDF。
The probability of an interval [a, b] can then be expressed as F(b) − F(a), regardless of whether the PDF is directly used.
于是区间 [a, b] 的概率可以表示为 F(b) − F(a)，而无需直接使用 PDF。
6. Expectation and Mean | 期望与均值
erage of all possible values, with weights given by the PDF. It is defined as E(X) = μ = ∫−∞∞ x f(x) dx. The expectation represents the centre of mass of the distribution and is a measure of central tendency.
连续随机变量 X 的期望值（或均值）是所有可能值的加权平均，权重由 PDF 给出。其定义为 E(X) = μ = ∫−∞∞ x f(x) dx。期望代表了分布的质心，是集中趋势的一种度量。
7. Variance and Standard Deviation | 方差与标准差
a distribution around its mean. For a continuous random variable, Var(X) = E[(X − μ)²] = ∫−∞∞ e square root of the variance, σ = √Var(X), and has the same units as the random variable.
方差衡量分布围绕其均值的扩散程度。对于连续随机变量，Var(X) = E[(X − μ)²] = ∫−∞∞ (x − μ)² f(x) dx。一个有用的计算形式是 Var(X) = E(X²) − [E(X)]²。标准差是方差的正平方根，σ = √Var(X)，并与随机变量具有相同的单位。
8. Continuous Uniform Distribution | 连续均匀分布
wise. This means every sub-interval of equal length has the same probability. The mean is (a + b)/2 and the variance is (b − a)²/12.
最简单的连续分布是区间 [a, b] 上的均匀分布。其 PDF 为常数：当 a ≤ x ≤ b 时 f(x) = 1/(b − a)，其余地方为 0。这意味着每个等长子区间具有相同的概率。均值为 (a + b)/2，方差为 (b − a)²/12。
≤ b, and F(x) = 1 for x > b. This linear CDF is easy to handle in probabilistic reasoning.
均匀分布的 CDF 为：x < a 时 F(x) = 0；a ≤ x ≤ b 时 F(x) = (x − a)/(b − a)；x > b 时 F(x) = 1。这种线性的 CDF 在概率推理中非常容易处理。
9. Median, Mode and Percentiles | 中位数、众数和百分位数
he area under the PDF into two equal halves. The mode is the value of x that maximises f(x); a distribution can be unimodal or multimodal. Percentiles are generalisations: the 100p-th percentile satisfies F(ηp) = p.
连续分布的中位数 m 是满足 P(X ≤ m) = 0.5 的值，即 F(m) = 0.5。它将 PDF 下的面积分为相等的两半。众数是使得 f(x) 最大的 x 值；分布可以是单峰的或多峰的。百分位数是中位数的推广：第 100p 百分位数满足 F(ηp) = p。
10. Normal Distribution | 正态分布
variable X that follows a normal distribution with mean μ and variance σ² is written as X ~ N(μ, σ²). Its PDF is the bell-shaped curve f(x) = (1/(σ√2π)) e−(x−μ)²/(2σ²). The curve is symmetric about μ and the spread is determined by σ.
正态分布是统计学中最重要的连续分布。若随机变量 X 服从均值为 μ、方差为 σ² 的正态分布，则记作 X ~ N(μ, σ²)。其 PDF 为钟形曲线 f(x) = (1/(σ√2π)) e−(x−μ)²/(2σ²)。该曲线关于 μ 对称，其离散程度由 σ 决定。
About 68% of the data lies within one standard deviation of the mean, 95% within two, and 99.7% within three. Many natural phenomena, measurement errors, and sample means are approximately normally distributed due to the Central Limit Theorem.
约 68% 的数据落在均值的一个标准差范围内，95% 落在两个标准差内，99.7% 落在三个标准差内。由于中心极限定理，许多自然现象、测量误差以及样本均值都近似服从正态分布。
11. Standard Normal Distribution and z-scores | 标准正态分布与z分数
The standard normal distribution is denoted Z ~ N(0, 1), with mean 0 and variance 1. Any normal variable X ~ ulting z-score indicates how many standard deviations a value is above or below the mean. Probabilities for any normal distribution can then be found using standard normal tables or a calculator.
标准正态分布记作 Z ~ N(0, 1)，其均值为 0，方差为 1。任何正态变量 X ~ N(μ, σ²) 都可以通过变换 Z = (X − μ)/σ 进行标准化。所得的 z 分数表明某个值高于或低于均值多少个标准差。随后可以利用标准正态分布表或计算器求得任何正态分布的概率。
12. Working with the Normal Distribution: Probability Calculations | 正态分布的概率计算
To find P(a < X < b) for X ~ thod is essential. Values such as Φ(−z) = 1 − Φ(z) simplify tail probability calculations.
对于 X ~ N(μ, σ²)，求 P(a < X < b) 时，首先将端点转换为 z 分数：z₁ = (a − μ)/σ, z₂ = (b − μ)/σ。于是 P(a < X < b) = P(z₁ < Z < z₂) = Φ(z₂) − Φ(z₁)，其中 Φ 是标准正态分布的 CDF。当代IB考试通常允许使用技术工具，但理解查表方法至关重要。诸如 Φ(−z) = 1 − Φ(z) 的关系能简化尾部概率的计算。
Inverse normal calculations – finding the value x for a given cumulative probability – t finds the z-score corresponding to the probability and then back-transforms: x = μ + zσ.
逆正态计算——在给定累积概率下求值 x——也很常见。此时先找到对应概率的 z 分数，然后逆变换：x = μ + zσ。
13. Sum of Independent Normal Variables | 独立正态变量之和
If X₁ ~ N(μ₁, σ₁²) and X₂ ~ N(μ₂, σ₂²) are independent, then their sum S = X₁ + X₂ is also normally distributed: S ~ l distribution and highly useful in problems involving multiple measurements or repeated sampling.
若 X₁ ~ N(μ₁, σ₁²) 和 X₂ ~ N(μ₂, σ₂²) 相互独立，则它们的和 S = X₁ + X₂ 也服从正态分布：S ~ N(μ₁ + μ₂, σ₁² + σ₂²)。更一般地，线性组合 aX₁ + bX₂ 服从 N(aμ₁ + bμ₂, a²σ₁² + b²σ₂²)。这种可加性是正态分布独有的性质，在处理涉及多次测量或重复抽样的问题时非常有用。
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