Continuous Random Variables | 连续型随机变量

📚 Continuous Random Variables | 连续型随机变量

A continuous random variable is one that can take any value within a given interval. Unlike discrete random variables, which are limited to isolated points, continuous variables arise from measurements—height, time, temperature, and so on. Understanding their behaviour requires the concepts of probability density functions, cumulative distributions, and the associated measures of centre and spread.

连续型随机变量可以在一个给定区间内取任意值。与只能取孤立点的离散型随机变量不同，连续变量来源于测量——如身高、时间、温度等。要理解它们的行为，需要掌握概率密度函数、累积分布函数以及相应的集中趋势和离散度量。

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

esents the height of an IB student, P(160 < X < 170) is meaningful, while P(X = 165) is 0.

连续型随机变量 X 可以在区间 [a, b] 甚至整个实数轴上取无限多个值。每一个单独结果的概率为 0，我们转而讨论 X 落在某个区间的概率。例如，若 X 表示 IB 学生的身高，P(160 < X < 170) 是有意义的，而 P(X = 165) = 0。

The graph of its probability model is a smooth curve called the probability density function (pdf). The total area under this curve equals 1, representing certainty that X takes some real value.

其概率模型的图像是一条光滑曲线，称为概率密度函数 (pdf)。曲线下的总面积等于 1，代表 X 必定取某个实数值。

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2. Probability Density Function (PDF) | 概率密度函数 (PDF)

sfies two conditions: f(x) ≥ 0 for all x, and the total area under the curve is 1, i.e., ∫_-∞^∞ f(x) dx = 1.

函数 f(x) 成为连续型随机变量 X 的有效 pdf 需满足两个条件：对所有 x，f(x) ≥ 0，且曲线下总面积为 1，即 ∫_-∞^∞ f(x) dx = 1。

two values a and b is given by the definite integral of f(x) from a to b: P(a ≤ X ≤ b) = ∫_a^b f(x) dx. Inclusion or exclusion of endpoints does not change the probability in a continuous setting.

X 落在 a 与 b 之间的概率由 f(x) 从 a 到 b 的定积分给出：P(a ≤ X ≤ b) = ∫_a^b f(x) dx。在连续情形中，包含或不包含端点并不改变概率。

A pdf can take values greater than 1, as long as the total area remains 1. The height f(x) itself does not directly give a probability—only areas do.

pdf 可以取大于 1 的值，只要总面积保持为 1。高度 f(x) 本身并不直接给出概率——只有面积表示概率。

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3. Cumulative Distribution Function (CDF) | 累积分布函数 (CDF)

n, denoted F(x), is defined as F(x) = P(X ≤ x). For a continuous random variable, F(x) = ∫_-∞^x f(t) dt. It represents the accumulated probability up to x.

累积分布函数记为 F(x)，定义为 F(x) = P(X ≤ x)。对连续型随机变量，F(x) = ∫_-∞^x f(t) dt。它表示到 x 为止的累积概率。

F(x) is a non-decreasing function with limits: lim_x→-∞ F(x) = 0 and lim_x→∞ differentiable wherever f(x) is continuous, and its derivative recovers the pdf: f(x) = F'(x).

F(x) 是一个非降函数，满足极限：lim_x→-∞ F(x) = 0，lim_x→∞ F(x) = 1。它在 f(x) 连续处连续且可导，其导数可以还原 pdf：f(x) = F'(x)。

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4. Relationship Between PDF and CDF | PDF 与 CDF 的关系

Differentiation and integration link the two functions: if you know the pdf, you integrate to get the cdf; if you have the cdf, you differentiate to obtain the pdf. This mirrors the relationship between velocity and displacement.

微分与积分将这两个函数联系起来：已知 pdf 时，积分得到 cdf；已知 cdf 时，求导便得 pdf。这与速度与位移的关系类似。

Probabilities can be expressed using either function: P(a ≤ X ≤ b) = F(b) − F(a) = ∫_a^b f(x) dx. When the pdf is given by a piecewise function, the cdf must be built piecewise as well, ensuring continuity.

概率可以用任一函数表示：P(a ≤ X ≤ b) = F(b) − F(a) = ∫_a^b f(x) dx。当 pdf 以分段函数给出时，cdf 也必须分段构造，并确保连续性。

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5. Expected Value (Mean) | 期望值（均值）

r mean, of X is a measure of its central tendency. For a continuous variable, E(X) = μ = ∫_-∞^∞ x f(x) dx. It locates the balance point of the distribution.

期望值或均值是 X 集中趋势的度量。对连续变量，E(X) = μ = ∫_-∞^∞ x f(x) dx。它确定了分布的“平衡点”位置。

d the integral converges. The expected value of a function g(X) is computed as E[g(X)] = ∫_-∞^∞ g(x) f(x) dx, which is particularly useful for finding E(X²).

若 pdf 关于点 c 对称，则 E(X) = c（假设积分收敛）。一个函数 g(X) 的期望值计算为 E[g(X)] = ∫_-∞^∞ g(x) f(x) dx，这个公式对求 E(X²) 特别有用。

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6. Variance and Standard Deviation | 方差与标准差

The variance measures the spread of X around its mean: Var(X) = E[(X − μ)²] = ∫_-∞^∞ In practice, the shortcut formula Var(X) = E(X²) − [E(X)]² often simplifies calculations.

方差衡量 X 围绕其均值的分散程度：Var(X) = E[(X − μ)²] = ∫_-∞^∞ (x − μ)² f(x) dx。实际中，简便公式 Var(X) = E(X²) − [E(X)]² 常能简化计算。

f the variance: σ = √Var(X). It shares the same units as X, making it easier to interpret.

标准差是方差的正平方根：σ = √Var(X)。它与 X 的单位相同，更易于解释。

tion Y = aX + b, the variance and standard deviation transform as Var(Y) = a² Var(X) and σ_Y = |a| σ_X, while E(Y) = a E(X) + b.

对于线性变换 Y = aX + b，方差和标准差的变换为 Var(Y) = a² Var(X)，σ_Y = |a| σ_X，而 E(Y) = a E(X) + b。

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7. Mode and Median | 众数与中位数

The mode of a continuous distribution is the value(s) of x that maximise the pdf f(x). A distribution can be unimodal (one peak), bimodal, or multimodal.

连续分布的众数是令 pdf f(x) 取得最大值的 x。分布可以是单峰的、双峰的或多峰的。

The median m is the value that splits the total area into two equal halves, satisfying ∫_-∞^m equivalently F(m) = 0.5. It is a robust centre measure, little affected by extreme values.

中位数 m 是将总面积分为相等两半的值，满足 ∫_-∞^m f(x) dx = 0.5 或等价于 F(m) = 0.5。它是一个稳健的中心度量，受极端值影响很小。

In symmetric uni-modal distributions, the mean, median, and mode all coincide. In skewed distributions they differ and their positions help describe the shape.

在对称单峰分布中，均值、中位数和众数重合。在偏态分布中它们各不相同，其相对位置有助于描述分布形状。

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8. The Continuous Uniform Distribution | 连续均匀分布

. This models situations where every subinterval of equal length has the same probability.

若随机变量 X 在区间 [a, b] 上具有恒定的 pdf：对于 a ≤ x ≤ b，f(x) = 1/(b − a)，其余为 0，那么 X 服从连续均匀分布。它适用于每个等长子区间具有相同概率的情形。

nd Var(X) = (b − a)²/12. The cdf is a linear ramp: F(x) = (x − a)/(b − a) for x in [a, b].

其均值和方差很容易得出：E(X) = (a + b)/2，Var(X) = (b − a)²/12。CDF 是一条线性斜线：当 x 在 [a, b] 时，F(x) = (x − a)/(b − a)。

The uniform distribution is commonly used in simulations, random number generation, and as a non-informative prior in statistical inference.

均匀分布常用于模拟、随机数生成以及作为统计推断中的无信息先验。

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9. The Normal Distribution | 正态分布

The normal distribution, famously bell-shaped, is central to statistics and the IB syllabus. A normal random variable X~N(μ, σ²) has pdf:

f(x) = (1/(σ√2π)) e^{−(x−μ)²/(2σ²)}, x ∈ ℝ

正态分布呈著名的钟形，是统计学和 IB 课程的核心。正态随机变量 X~N(μ, σ²) 的 pdf 为上述公式。

ymmetric about x = μ, where it reaches its maximum; the points of inflection are at μ ± σ.

参数 μ 与 σ 分别代表均值和标准差。曲线关于 x = μ 对称，并在该处达最大值；拐点位于 μ ± σ 处。

Important properties: about 68% of data lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. The distribution extends infinitely in both tails, but the total area is 1.

重要性质：大约 68% 的数据落在均值 ±1σ 内，95% 在 ±2σ 内，99.7% 在 ±3σ 内。分布向两侧无限延伸，总面积却为 1。

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10. Standard Normal Distribution and Z‑Scores | 标准正态分布与 Z 分数

Any normal variable can be standardised by subtracting μ and dividing by σ. The Z‑score Z = (X − μ)/σ follows the standard normal distribution N(0, 1) with pdf φ(z) = (1/√2π) e^−z²/2. Its mean is 0 and standard deviation 1.

任何一个正态变量都可以通过减去 μ 并除以 σ 实现标准化。Z 分数 Z = (X − μ)/σ 服从标准正态分布 N(0, 1)，其 pdf 为 φ(z) = (1/√2π) e^−z²/2，均值为 0，标准差为 1。

ilities for any normal distribution can be converted to standard normal probabilities: P(X ≤ k) = P(Z ≤ (k − μ)/σ). Tables or technology then give the required tail areas.

任意正态分布的概率都可以转化为标准正态概率：P(X ≤ k) = P(Z ≤ (k − μ)/σ)。然后可借助表格或技术查找所需的尾部面积。

Often you will be asked to work backwards: given a probability, find the corresponding x‑value using the inverse normal function on a GDC or inverse lookup in Z‑tables.

经常需要反向求解：给定概率，使用 GDC 上的逆正态函数或 Z 表逆查找到对应的 x 值。

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11. Using Technology to Find Probabilities | 使用技术求概率

In IB examinations, a graphical display calculator (GDC) is an indispensable tool. You can calculate normal probabilities directly using the 'normalcdf' function without z‑score conversion. For example, on TI‑Nspire or Casio, normalcdf(lower, upper, μ, σ) gives P(lower < X < upper).

在 IB 考试中，图形计算器 (GDC) 是不可或缺的工具。你可以直接使用 'normalcdf' 函数求正态概率，无需转换 z 分数。例如，在 TI‑Nspire 或 Casio 计算器上，normalcdf(下限, 上限, μ, σ) 给出 P(下限 < X < 上限)。

The 'invNorm' function solves the reverse problem: invNorm(area, μ, σ) returns the value x such that P(X < x) equals the specified area (left‑tail). When only standard normal tables are available, you must first convert to Z scores and then interpolate if necessary.

'invNorm' 函数解决逆向问题：invNorm(面积, μ, σ) 返回使 P(X < x) 等于给定左侧面积的值 x。如果只有标准正态表，必须先转换为 Z 分数，必要时还需进行插值。

For a continuous uniform distribution, probabilities, means, and variances are usually computed analytically; however, GDCs can be used to check definite integrals or to sketch the pdf.

对于连续均匀分布，概率、均值和方差通常通过解析方法计算；不过，也可以用 GDC 检验定积分或描绘 pdf 草图。

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12. Key Formulas Summary | 核心公式小结

The following table collects the fundamental expressions you are expected to know for the IB analysis.

下表汇总了 IB 考试中需要掌握的基本公式。

Concept	Formula	中文
Total probability	∫-∞∞ f(x) dx = 1	全概率为 1
Probability interval	P(a ≤ X ≤ b) = ∫ab f(x) dx = F(b) − F(a)	区间概率
Mean	μ = E(X) = ∫-∞∞ x f(x) dx	均值
Variance	Var(X) = E(X²) − μ² = ∫-∞∞ (x−μ)² f(x) dx	方差
Median	F(m) = 0.5, i.e. ∫-∞m f(x) dx = 0.5	中位数
Standardisation	Z = (X − μ)/σ ~ N(0,1)	标准化

Commit these to memory, and practice applying them in diverse contexts—from uniform to normal distributions. A solid grasp of the interplay between differentiation, integration, and probability will serve you well in Paper 2 and beyond.

请牢记这些公式，并练习在各种情境中应用它们——从均匀分布到正态分布。深刻理解微分、积分与概率之间的关系将对 Paper 2 及未来的学习大有裨益。

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