📚 IB Math: Distribution Properties of Continuous Random Variables | IB 数学：连续型随机变量的分布特征
A continuous random variable takes infinitely many possible values within an interval, and its behavior is described by a probability density function (pdf) rather than by individual probabilities. In IB Mathematics, understanding the key features of a continuous distribution – the pdf, cumulative distribution function (cdf), expectation, variance, median, and mode – is essential for solving problems in probability and statistics.
连续型随机变量在一个区间内可以取无限多个可能的值，其行为由概率密度函数（pdf）而不是单个概率来描述。在 IB 数学中，理解连续分布的关键特征——概率密度函数、累积分布函数（cdf）、期望、方差、中位数和众数——对于解决概率与统计问题至关重要。
1. Definition of a Continuous Random Variable | 连续型随机变量的定义
A random variable X is continuous if its possible values form an entire interval of real numbers. For example, the height of a randomly selected student, the time until a radioactive particle decays, or the exact weight of a product are all continuous random variables. Because there are infinitely many uncountable outcomes, the probability that X equals any specific single value is zero: P(X = a) = 0.
如果随机变量 X 的可能取值构成实数轴上的一个完整区间，则称 X 为连续型随机变量。例如，随机挑选一名学生的身高、放射性粒子衰变的时间、或一件产品的精确重量都是连续型随机变量。由于存在无限多个不可数结果，X 等于某个特定值的概率为零：P(X = a) = 0。
This key idea leads to a useful consequence: for continuous variables, inequalities do not distinguish between strict and inclusive endpoints. Thus P(a ≤ X ≤ b) = P(a < X < b).
这一关键思想引出了一个有用的结论：对于连续变量，不等式中严格与包含端点没有区别。因此 P(a ≤ X ≤ b) = P(a < X < b)。
2. Probability Density Function (pdf) | 概率密度函数
The probability density function, denoted f(x), must satisfy two conditions for all real x: f(x) ≥ 0, and the total area under the curve is 1, so ∫ f(x) dx from -∞ to ∞ = 1. The probability that X lies between a and b is the area under the curve from a to b: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx.
概率密度函数记为 f(x)，它必须对所有实数 x 满足两个条件：f(x) ≥ 0，且曲线下方总面积为 1，即从 -∞ 到 ∞ 的积分 ∫ f(x) dx = 1。X 位于 a 和 b 之间的概率是曲线从 a 到 b 下方的面积：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。
Because P(X = a) = 0, the height f(a) itself is not a probability. It can be greater than 1, provided the total area is 1. The pdf gives the relative likelihood of values near a: the higher f(a), the more likely X is to fall in a small interval around a.
因为 P(X = a) = 0，所以 f(a) 本身不是概率。它可以大于 1，只要总面积仍为 1。pdf 给出了 a 附近值的相对似然：f(a) 越大，X 落入 a 附近的小区间的可能性就越大。
3. Cumulative Distribution Function (cdf) | 累积分布函数
The cumulative distribution function F(x) is defined as F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt. It gives the probability that X takes a value less than or equal to x. F(x) is a non-decreasing function with limits 0 as x → -∞ and 1 as x → ∞.
累积分布函数定义为 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt，它表示 X 取值小于或等于 x 的概率。F(x) 是非递减函数，当 x → -∞ 时极限为 0，当 x → ∞ 时极限为 1。
Using the cdf, probabilities can be computed as differences: P(a < X ≤ b) = F(b) - F(a). Moreover, if the pdf is continuous at x, then F'(x) = f(x). In IB problems, you may need to find F(x) from f(x) by integration, or find f(x) from F(x) by differentiation.
利用 cdf，概率可以通过差值计算：P(a < X ≤ b) = F(b) - F(a)。此外，如果 pdf 在 x 处连续，则 F'(x) = f(x)。在 IB 题目中，你可能需要通过积分由 f(x) 求 F(x)，或通过求导由 F(x) 求 f(x)。
4. Expectation E(X) | 期望 E(X)
The expected value (mean) of a continuous random variable is the weighted average of all possible values, weighted by the density: E(X) = ∫₋∞∞ x f(x) dx. It represents the long-run average value of X in many trials and may not be a possible value of X itself.
连续型随机变量的期望值（均值）是所有可能值以密度为权重的加权平均：E(X) = ∫₋∞∞ x f(x) dx。它表示多次试验中 X 的长期平均值，本身可能不是 X 的可能取值。
For any function g(X), the expectation is E[g(X)] = ∫₋∞∞ g(x) f(x) dx. In particular, E(aX + b) = aE(X) + b, which is useful when transforming variables.
对于任意函数 g(X)，期望为 E[g(X)] = ∫₋∞∞ g(x) f(x) dx。特别地，E(aX + b) = aE(X) + b，这在变量变换时很有用。
E(X) = ∫₋∞∞ x f(x) dx
5. Variance and Standard Deviation | 方差与标准差
Variance measures the spread of a distribution around its mean. It is defined as Var(X) = ∫₋∞∞ (x – μ)² f(x) dx, where μ = E(X). The standard deviation is σ = √Var(X).
方差度量分布围绕其均值的离散程度，定义为 Var(X) = ∫₋∞∞ (x – μ)² f(x) dx，其中 μ = E(X)。标准差为 σ = √Var(X)。
A more convenient computational formula is Var(X) = E(X²) – [E(X)]². Since E(X²) = ∫₋∞∞ x² f(x) dx, this often simplifies calculations. Also, Var(aX + b) = a²Var(X), showing that adding a constant does not affect spread, while multiplying by a scales it quadratically.
更便捷的计算公式是 Var(X) = E(X²) – [E(X)]²。因为 E(X²) = ∫₋∞∞ x² f(x) dx，这常能简化计算。同时，Var(aX + b) = a²Var(X)，表明加常数不影响离散程度，而乘以常数会按平方缩放离散程度。
6. Median and Quartiles | 中位数与四分位数
The median m satisfies F(m) = 0.5, meaning exactly half of the probability lies below m and half above. For a distribution with pdf f(x), the median is found by solving ∫₋∞ᵐ f(x) dx = 0.5.
中位数 m 满足 F(m) = 0.5，即刚好一半的概率位于 m 以下，一半位于 m 以上。对于具有 pdf f(x) 的分布，中位数通过求解 ∫₋∞ᵐ f(x) dx = 0.5 得到。
Similarly, the lower quartile Q1 solves F(Q1) = 0.25, and the upper quartile Q3 solves F(Q3) = 0.75. The interquartile range (IQR) is Q3 – Q1 and provides a robust measure of spread that is less affected by outliers.
类似地，下四分位数 Q1 满足 F(Q1) = 0.25，上四分位数 Q3 满足 F(Q3) = 0.75。四分位距（IQR）为 Q3 – Q1，是受异常值影响较小的离散程度度量。
7. Mode | 众数
The mode of a continuous distribution is the value of x where the pdf f(x) attains its maximum. It represents the most likely value in a small interval around that point. A distribution can have one mode (unimodal), two modes (bimodal), or many modes.
连续分布的众数是 pdf f(x) 取得最大值的 x 值，它表示该点附近小区间内最可能出现的值。分布可以有一个众数（单峰）、两个众数（双峰）或多个众数。
To find the mode analytically, set f'(x) = 0 and verify that f”(x) < 0, or check the endpoints of the support. For example, a normal distribution has a unique mode at its mean μ.
解析求众数时，令 f'(x) = 0 并验证 f”(x) < 0，或检查定义域的端点。例如，正态分布在其均值 μ 处有唯一的众数。
8. Uniform Distribution | 均匀分布
The continuous uniform distribution on [a, b] has a constant pdf f(x) = 1/(b – a) for x in [a, b] and 0 elsewhere. Because the density is constant, all intervals of equal length have equal probability.
在 [a, b] 上的连续均匀分布具有常数 pdf：当 x 在 [a, b] 内时 f(x) = 1/(b – a)，其余为 0。由于密度恒定，所有等长区间具有相等的概率。
Its cdf is F(x) = 0 for x < a, F(x) = (x - a)/(b - a) for a ≤ x ≤ b, and F(x) = 1 for x > b. The mean is E(X) = (a + b)/2, the midpoint, and the variance is Var(X) = (b – a)²/12.
其 cdf 为：x < a 时 F(x) = 0，a ≤ x ≤ b 时 F(x) = (x - a)/(b - a)，x > b 时 F(x) = 1。均值为 E(X) = (a + b)/2，即中点；方差为 Var(X) = (b – a)²/12。
9. Normal Distribution | 正态分布
The normal distribution N(μ, σ²) is the most important continuous distribution in statistics. Its pdf is symmetric about the mean μ and bell-shaped: f(x) = 1/(σ√(2π)) · e^[-(x-μ)²/(2σ²)].
正态分布 N(μ, σ²) 是统计学中最重要的连续分布。其 pdf 关于均值 μ 对称，呈钟形：f(x) = 1/(σ√(2π)) · e^[-(x-μ)²/(2σ²)]。
The standard normal distribution Z has μ = 0 and σ = 1. Any normal variable X can be standardized by Z = (X – μ)/σ. IB questions usually require using the standard normal table or a GDC to find probabilities such as P(Z < 1.5) or P(-1.2 < Z < 0.8).
标准正态分布 Z 具有 μ = 0，σ = 1。任何正态变量 X 都可以通过 Z = (X – μ)/σ 进行标准化。IB 题目通常要求使用标准正态分布表或计算器（GDC）求概率，如 P(Z < 1.5) 或 P(-1.2 < Z < 0.8)。
Z = (X – μ)/σ
10. Exponential Distribution | 指数分布
The exponential distribution is often used to model waiting times or lifetimes. Its pdf is f(x) = λe^(-λx) for x ≥ 0, where λ > 0 is the rate parameter. The cdf is F(x) = 1 – e^(-λx) for x ≥ 0.
指数分布常用于建模等待时间或寿命。其 pdf 为当 x ≥ 0 时 f(x) = λe^(-λx)，其中 λ > 0 是速率参数。cdf 为当 x ≥ 0 时 F(x) = 1 – e^(-λx)。
The mean and standard deviation are both 1/λ, so E(X) = 1/λ and Var(X) = 1/λ². An important property is memorylessness: P(X > s + t | X > s) = P(X > t), meaning the remaining lifetime does not depend on how long one has already waited.
指数分布的均值和标准差都为 1/λ，即 E(X) = 1/λ，Var(X) = 1/λ²。一个重要性质是无记忆性：P(X > s + t | X > s) = P(X > t)，即剩余寿命与已等待时间无关。
11. Properties and Applications in IB | 性质与 IB 应用
When tackling continuous random variable problems in IB, first check whether the given function is a valid pdf by verifying non-negativity and total area equal to 1. Then, for probabilities, integrate the pdf over the required interval. For expectations, integrate x·f(x) or x²·f(x) as needed.
在 IB 中处理连续型随机变量问题时，首先检查给定函数是否为有效 pdf，即验证非负性和总面积等于 1。然后，求概率时在所需区间上对 pdf 积分；求期望时根据需要积分 x·f(x) 或 x²·f(x)。
Common pitfalls include forgetting to adjust integration bounds when the support is restricted, confusing pdf values with probabilities, and incorrectly computing E(X²) from E(X). Always sketch the density curve if possible to visualize areas representing probabilities.
常见错误包括：定义域受限时忘记调整积分限；将 pdf 值误认为概率；以及从 E(X) 错误地计算 E(X²)。如果可能，务必画出密度曲线，以可视化表示概率的面积。
12. Exam Tips and Summary | 考试提示与总结
In the IB exam, you should be comfortable using a GDC to compute normal probabilities and inverse normal values. For other continuous distributions, show the integral setup clearly and use exact values where required. Always state units and round final answers to the appropriate degree of accuracy.
在 IB 考试中，你应熟练使用 GDC 计算正态概率和逆正态值。对于其他连续分布，要清晰写出积分表达式，并在需要时使用精确值。始终注明单位，并按要求的精度保留最终答案。
To summarize, the distribution of a continuous random variable is fully characterized by its pdf f(x) and cdf F(x). Expectation, variance, median, and mode are derived from these functions and provide measures of centre and spread. Mastery of these concepts allows you to analyze real-world continuous data confidently.
总结而言，连续型随机变量的分布完全由其 pdf f(x) 和 cdf F(x) 刻画。期望、方差、中位数和众数都从这些函数导出，提供了集中趋势和离散程度的度量。掌握这些概念，你就能自信地分析真实世界中的连续数据。
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