📚 Continuous Probability Distributions | 连续概率分布
Continuous probability distributions form the backbone of statistical analysis in the IB Mathematics curriculum. They allow us to model and predict the behaviour of random variables that can take any value within a certain interval. Understanding these distributions is essential for performing hypothesis tests, constructing confidence intervals, and interpreting real-world data from natural and social sciences.
连续概率分布是IB数学课程中统计分析的基石。它们使我们能够对在某一区间内可以取任意值的随机变量进行建模和预测。理解这些分布对于进行假设检验、构建置信区间以及解释来自自然科学和社会科学的现实世界数据至关重要。
1. Introduction to Continuous Distributions | 连续分布简介
A continuous random variable can assume any value within a given range. Unlike discrete variables, which have distinct outcomes, continuous variables are measured rather than counted – think of height, time, or temperature. The probability of the variable taking any exact single value is zero; we always talk about intervals.
连续型随机变量可以在给定范围内取任意值。与具有独立结果的离散变量不同，连续变量是测量得来的而不是计数得来的——想想身高、时间或温度。该变量取任何一个精确单值的概率为零；我们总是讨论区间内的概率。
In IB Mathematics, you will encounter continuous distributions to quantify uncertainty. The total area under the probability density curve represents the entire sample space and equals 1. This concept replaces the probability mass function used for discrete distributions.
在IB数学中，您将会遇到连续分布以量化不确定性。概率密度曲线下的总面积代表整个样本空间，等于1。这个概念取代了离散分布中使用的概率质量函数。
2. Probability Density Function (PDF) | 概率密度函数
The probability density function, written as f(x), describes the relative likelihood of a continuous random variable occurring near a point. The key idea is that probability is given by the area under the curve: P(a ≤ X ≤ b) = ∫ab f(x) dx.
概率密度函数，记作 f(x)，描述了一个连续随机变量在某个点附近发生的相对可能性。其核心思想是：概率由曲线下的面积给出：P(a ≤ X ≤ b) = ∫ab f(x) dx。
For any continuous random variable, f(x) ≥ 0 for all x, and the integral over the entire domain equals 1. Note that f(x) itself is not a probability – it can be greater than 1, provided the total area integrates to 1.
对于任何连续随机变量，对所有 x 都有 f(x) ≥ 0，并且在整个定义域上的积分等于1。注意 f(x) 本身不是概率——它可以大于1，只要总面积积分为1即可。
The graph of a PDF is a smooth curve. The mode of the distribution occurs at the x-value where f(x) is maximised, while the median splits the area into two equal halves.
概率密度函数的图像是一条平滑曲线。分布的众数出现在 f(x) 取最大值的 x 处，而中位数将面积分成相等的两部分。
3. Cumulative Distribution Function (CDF) | 累积分布函数
The cumulative distribution function, denoted F(x), gives the probability that the random variable X is less than or equal to x: F(x) = P(X ≤ x) = ∫-∞x f(t) dt. It always starts at 0, ends at 1, and is non-decreasing.
累积分布函数，记作 F(x)，给出了随机变量 X 小于或等于 x 的概率：F(x) = P(X ≤ x) = ∫-∞x f(t) dt。它总是从 0 开始，到 1 结束，并且是非递减的。
One crucial relationship is that the derivative of the CDF gives back the PDF: f(x) = F'(x). This means we can move freely between the two functions. To find P(a < X < b), we simply compute F(b) − F(a).
一个至关重要的关系是：累积分布函数的导数就是概率密度函数：f(x) = F'(x)。这意味着我们可以在这两个函数之间自由转换。要计算 P(a < X < b)，我们只需计算 F(b) − F(a)。
For many continuous models, the CDF is easier to work with because it directly yields probabilities for intervals without further integration. IB exams often provide CDF formulas or require you to derive them from a given PDF.
对于许多连续模型，累积分布函数更易于使用，因为它可以直接得出区间的概率而无需进一步积分。IB考试经常会提供累积分布函数公式，或者要求你根据给定的概率密度函数推导出来。
4. Properties of PDFs | 概率密度函数的性质
To verify a function is a valid PDF, two conditions must hold: f(x) ≥ 0 for all real x, and the total area ∫-∞∞ f(x) dx = 1. These conditions ensure that probabilities are never negative and the sum of all probabilities is 1.
要验证一个函数是否为有效的概率密度函数，必须满足两个条件：对所有实数 x，f(x) ≥ 0；并且总面积 ∫-∞∞ f(x) dx = 1。这些条件确保了概率永远不会为负，且所有概率之和为1。
Because the probability of a single point is zero for continuous variables, we have P(a ≤ X ≤ b) = P(a < X < b) = P(a ≤ X < b) = P(a < X ≤ b). This symmetry simplifies many calculations.
由于连续变量在单点处的概率为零，因此我们有 P(a ≤ X ≤ b) = P(a < X < b) = P(a ≤ X < b) = P(a < X ≤ b)。这种对称性简化了许多计算。
Furthermore, the area of a tail probability can be found either by direct integration or by subtracting the CDF value from 1: P(X > x) = 1 − F(x). This is especially useful when dealing with normal and exponential tail probabilities.
此外，尾部概率的面积可以通过直接积分或从1中减去累积分布函数值来求得：P(X > x) = 1 − F(x)。在处理正态分布和指数分布的尾部概率时，这个方法尤其有用。
5. Expected Value and Variance | 期望值与方差
The expected value (or mean) of a continuous random variable X is defined as μ = E(X) = ∫-∞∞ x f(x) dx. This represents the long-run average or centre of mass of the distribution.
连续随机变量 X 的期望值（或均值）定义为 μ = E(X) = ∫-∞∞ x f(x) dx。它代表了分布的长期平均值或质心。
The variance is Var(X) = E[(X − μ)²] = ∫-∞∞ (x − μ)² f(x) dx, often calculated using the shortcut formula Var(X) = E(X²) − [E(X)]², where E(X²) = ∫ x² f(x) dx. The standard deviation is σ = √Var(X).
方差为 Var(X) = E[(X − μ)²] = ∫-∞∞ (x − μ)² f(x) dx，通常使用简便公式 Var(X) = E(X²) − [E(X)]² 来计算，其中 E(X²) = ∫ x² f(x) dx。标准差为 σ = √Var(X)。
These parameters summarise the central tendency and the spread of the distribution. In the IB syllabus, you are expected to calculate expectations and variances for uniform, normal, and exponential distributions from their PDFs.
这些参数概括了分布的中心趋势和离散程度。在IB课程大纲中，要求你能够从均匀分布、正态分布和指数分布的概率密度函数中计算出期望和方差。
6. Uniform Distribution | 均匀分布
The continuous uniform distribution on the interval [a, b] has a constant probability density: f(x) = 1/(b − a) for a ≤ x ≤ b, and 0 elsewhere. Every sub-interval of equal length within [a, b] is equally likely.
区间 [a, b] 上的连续均匀分布具有恒定的概率密度：当 a ≤ x ≤ b 时，f(x) = 1/(b − a)；其他情况下为0。在 [a, b] 内，任何长度相等的子区间都同样可能。
The CDF of the uniform distribution is linear: F(x) = (x − a)/(b − a) for a ≤ x ≤ b. The mean is (a + b)/2 and the variance is (b − a)²/12. It is often used to model complete randomness, such as a random number generator.
均匀分布的累积分布函数是线性的：当 a ≤ x ≤ b 时，F(x) = (x − a)/(b − a)。均值为 (a + b)/2，方差为 (b − a)²/12。它通常用于对完全随机性建模，例如随机数生成器。
When solving problems, remember that for the uniform distribution, probabilities are simply lengths of intervals divided by total length: P(c ≤ X ≤ d) = (d − c)/(b − a). This makes it one of the easiest continuous distributions to handle.
在解决问题时，请记住对于均匀分布，概率就是区间长度除以总长度：P(c ≤ X ≤ d) = (d − c)/(b − a)。这使得它成为最容易处理的连续分布之一。
7. Normal Distribution | 正态分布
The normal distribution is the most important continuous distribution in statistics. Its PDF is bell-shaped and symmetric about the mean μ, with the spread determined by the standard deviation σ. The notation is X ~ N(μ, σ²).
正态分布是统计学中最重要的连续分布。其概率密度函数呈钟形，以均值 μ 为中心对称，分散程度由标准差 σ 决定。记作 X ~ N(μ, σ²)。
The PDF of a normal distribution is f(x) = 1/(σ √(2π)) e-(x-μ)²/(2σ²). While this formula looks intimidating, IB questions rarely require integration by hand; instead, you use the standard normal table or calculator functions.
正态分布的概率密度函数为 f(x) = 1/(σ √(2π)) e-(x-μ)²/(2σ²)。虽然这个公式看起来令人生畏，但IB考题很少要求手动积分；相反，你使用标准正态表或计算器功能。
The empirical rule (68–95–99.7 rule) states that approximately 68% of data fall within 1σ of μ, 95% within 2σ, and 99.7% within 3σ. This is a quick way to estimate probabilities for symmetric intervals.
经验法则（68–95–99.7法则）指出：大约68%的数据落在 μ 的 ±1σ 范围内，95%落在 ±2σ 范围内，99.7%落在 ±3σ 范围内。这是估计对称区间概率的快捷方法。
8. Standard Normal Distribution | 标准正态分布
The standard normal distribution is a special case with mean 0 and variance 1: Z ~ N(0, 1). Any normal variable can be standardised using the formula Z = (X − μ)/σ. Standardisation converts a normal value into a z-score, which tells how many standard deviations it is from the mean.
标准正态分布是均值为0、方差为1的特例：Z ~ N(0, 1)。任何正态变量都可以通过公式 Z = (X − μ)/σ 进行标准化。标准化将正态值转化为 z 分数，表示它距离均值多少个标准差。
The standard normal CDF, denoted Φ(z) = P(Z ≤ z), is tabulated or programmed into GDC calculators. To find P(a < X < b) for X ~ N(μ, σ²), compute Φ((b−μ)/σ) − Φ((a−μ)/σ).
标准正态累积分布函数记作 Φ(z) = P(Z ≤ z)，已制成表格并编入GDC计算器程序。对于 X ~ N(μ, σ²)，为求 P(a < X < b)，计算 Φ((b−μ)/σ) − Φ((a−μ)/σ)。
IB students should be comfortable both using technology and reading standard normal tables. Common values to remember: Φ(0) = 0.5, Φ(1) ≈ 0.8413, Φ(1.96) ≈ 0.975, which gives the famous 95% confidence z-score.
IB 学生应当既能够熟练运用技术，也能阅读标准正态表。需要记住的常用值有：Φ(0) = 0.5，Φ(1) ≈ 0.8413，Φ(1.96) ≈ 0.975，后者给出了著名的95%置信度的 z 分数。
9. Exponential Distribution | 指数分布
The exponential distribution models the time between events in a Poisson process, such as the lifetime of an electronic component or waiting time until the next customer arrives. Its PDF is f(x) = λ e-λx for x ≥ 0, where λ > 0 is the rate parameter.
指数分布模拟泊松过程中事件之间的时间，例如电子元件的寿命或下一个顾客到达前的等待时间。其概率密度函数为当 x ≥ 0 时 f(x) = λ e-λx，其中 λ > 0 是速率参数。
The CDF is easily integrable: F(x) = 1 − e-λx for x ≥ 0. The mean is 1/λ and the variance is 1/λ². Because of the memoryless property, P(X > t + s | X > t) = P(X > s), which makes it unique among continuous distributions.
累积分布函数易于积分：当 x ≥ 0 时 F(x) = 1 − e-λx。均值为 1/λ，方差为 1/λ²。由于无记忆性，P(X > t + s | X > t) = P(X > s)，这使得它在连续分布中独一无二。
In IB problems, you might be asked to find the median (solve F(m) = 0.5) or to determine λ from a given mean. Always check that the variable is defined only for non-negative x.
在IB题目中，可能会要求求出中位数（解 F(m) = 0.5）或根据给定的均值确定 λ。务必检查变量是否只对 x ≥ 0 定义。
10. Using the CDF to Find Probabilities | 使用累积分布函数求概率
Instead of integrating a PDF each time, we can directly use the CDF to find interval probabilities. For a continuous random variable X, P(X ≤ a) = F(a); P(X > a) = 1 − F(a); and P(a < X < b) = F(b) − F(a).
我们无需每次都积分概率密度函数，而是可以直接使用累积分布函数来计算区间概率。对于连续随机变量 X，P(X ≤ a) = F(a)；P(X > a) = 1 − F(a)；且 P(a < X < b) = F(b) − F(a)。
These relationships are universal. They hold regardless of the shape of the distribution. The CDF is therefore a powerful tool: once you know F(x), you can answer most probability questions without performing any integration.
这些关系是普适的。无论分布形状如何，它们都成立。因此累积分布函数是一个强大的工具：一旦你知道 F(x)，就可以在无需进行任何积分的情况下回答大多数概率问题。
Even when a PDF has multiple pieces, the CDF can be built by carefully accumulating area. IB papers will sometimes provide a piecewise PDF and require you to construct F(x) over the domain.
即使概率密度函数有多段，累积分布函数也可以通过仔细累积面积来构建。IB试卷有时会提供一个分段概率密度函数，并要求你在定义域上构建 F(x)。
11. Transformations and Standardization | 变换与标准化
Transforming a continuous random variable changes its distribution. A linear transformation Y = aX + b has mean E(Y) = aμ + b and variance Var(Y) = a²σ². This is especially relevant when converting between normal distributions and the standard normal.
变换一个连续随机变量会改变其分布。线性变换 Y = aX + b 的均值为 E(Y) = aμ + b，方差为 Var(Y) = a²σ²。这在正态分布与标准正态分布之间转换时尤其相关。
Standardisation is a specific linear transformation: Z = (X − μ)/σ. It centres the variable at 0 and scales the spread to 1. Once in standard form, we can use pre-computed tables or calculator functions to find probabilities.
标准化是一种特定的线性变换：Z = (X − μ)/σ。它将变量中心移到 0，并将离散程度缩放到 1。一旦转化为标准形式，我们就可以使用预先计算的表格或计算器功能来求概率。
Non-linear transformations (such as Y = X², Y = eX) are more involved and usually lie beyond the core IB syllabus, but the concept of expectation as an integral still applies to find mean and variance directly.
非线性变换（如 Y = X²，Y = eX）更为复杂，通常超出了IB核心课程的范围，但期望作为积分的概念仍然适用于直接求取均值和方差。
12. Applications and Summary | 应用与总结
Continuous distributions appear throughout science and engineering: normal distributions model measurement errors and biological traits, exponential distributions model radioactive decay and waiting times, and uniform distributions model fair random sampling.
连续分布贯穿于科学和工程学中：正态分布为测量误差和生物特征建模，指数分布为放射性衰变和等待时间建模，均匀分布为公平随机抽样建模。
In the IB exam, you may encounter problems that combine PDF and CDF concepts, require you to find unknown constants from total area = 1, and apply standardisation to calculate normal probabilities. Practising with real data sets enhances understanding.
在IB考试中，你可能会遇到结合概率密度函数和累积分布函数概念的题目，要求你通过总面积=1求出未知常数，并应用标准化来计算正态概率。使用真实数据集进行练习可以增强理解。
The core ideas to remember are: probabilities are areas, the total area is 1, the CDF gives cumulative probabilities, and standardisation brings any normal variable to a common scale. Mastering continuous distributions builds a solid foundation for further topics like statistical inference and regression.
需要记住的核心思想是：概率即面积，总面积为1，累积分布函数给出累计概率，标准化将任何正态变量带到同一尺度。掌握连续分布为进一步探讨统计推断和回归等课题奠定了坚实基础。
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