📚 Random Variables and Probability Distributions Overview | 随机变量与概率分布总览
Random variables and probability distributions form the cornerstone of statistical inference and probability theory in IB Mathematics. This section bridges the gap between purely theoretical probability and real-world data analysis, making it a high-yield topic for exams. A solid grasp of the definitions, properties, and standard distributions is essential for achieving top marks in the Paper 3 and the internal assessment.
随机变量与概率分布构成了 IB 数学中统计推断与概率论的基石。这一部分弥合了纯理论概率与现实世界数据分析之间的鸿沟，是考试中的高价值考点。扎实掌握定义、性质以及标准分布，对于在试卷三和内部评估中取得高分至关重要。
1. Discrete Random Variables | 离散随机变量
A discrete random variable (DRV) is a variable that can take on a finite or countably infinite set of distinct values. The probability of each specific value is defined by the probability mass function (PMF), denoted as P(X = x). The PMF must satisfy two fundamental conditions: each probability is non-negative, and the sum of all probabilities over the possible values equals one.
离散随机变量 (DRV) 是在有限或可数无穷个不同值上取值的变量。每个具体值的概率由概率质量函数 (PMF) 定义，记为 P(X = x)。PMF 必须满足两个基本条件：每个概率非负，且所有可能值上的概率之和等于一。
The expected value, E(X), represents the long-run average or the center of the distribution. The variance, Var(X), quantifies the spread or dispersion of the distribution around the mean. The standard deviation is the square root of the variance. These parameters allow us to summarize the entire distribution with a few meaningful numbers.
期望值 E(X) 代表长期平均值或分布的中心位置。方差 Var(X) 量化了分布围绕均值的离散程度或扩散程度。标准差是方差的平方根。这些参数使我们能够用几个有意义的数字来概括整个分布。
E(X) = Σ x · P(X = x) and Var(X) = Σ x² · P(X = x) – μ²
2. Continuous Random Variables | 连续随机变量
A continuous random variable (CRV) takes values over an interval or continuum of real numbers. Its distribution is described by a probability density function (PDF), f(x). Unlike the discrete case, the probability of observing any single exact point is zero; probabilities must always be calculated over an interval, representing the area under the PDF curve.
连续随机变量 (CRV) 在实数轴的区间或连续统上取值。其分布由概率密度函数 (PDF) f(x) 描述。与离散情形不同，观测到任意单个精确点的概率为零；概率必须始终在某个区间上计算，即 PDF 曲线下的面积。
The cumulative distribution function (CDF), F(x) = P(X ≤ x), is the integral of the PDF from negative infinity up to x. For a valid PDF, the total area under the curve must equal 1. The expected value is the integral of x times the density, and the variance is found using the second moment.
累积分布函数 (CDF) F(x) = P(X ≤ x) 是 PDF 从负无穷到 x 的积分。对于有效的 PDF，曲线下的总面积必须等于 1。期望值是 x 乘以密度的积分，方差通过二阶矩求得。
P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx, E(X) = ∫ x f(x) dx, Var(X) = ∫ x² f(x) dx – μ²
3. Binomial Distribution | 二项分布
The binomial distribution is one of the most important discrete probability models. It applies to a scenario with a fixed number of independent trials, n, where each trial has exactly two possible outcomes (success or failure) and the probability of success, p, is constant across all trials. The number of successes X is then denoted by X ~ B(n, p).
二项分布是最重要的离散概率模型之一。它适用于以下情景：固定次数的独立试验 n，每次试验恰好有两种可能结果（成功或失败），且成功概率 p 在所有试验中恒定。成功次数 X 记为 X ~ B(n, p)。
The PMF gives the probability of obtaining exactly x successes. The mean is np and the variance is np(1-p). The conditions for using the binomial distribution are crucial: a fixed number of trials, independence between trials, and a constant probability of success. When these conditions are violated, the distribution becomes inapplicable.
PMF 给出了恰好获得 x 次成功的概率。均值为 np，方差为 np(1-p)。使用二项分布的条件至关重要：固定次数的试验、试验间相互独立以及恒定的成功概率。当这些条件被违反时，该分布将不再适用。
P(X = x) = C(n, x) · pˣ · (1-p)ⁿ⁻ˣ, E(X) = np, Var(X) = np(1-p)
4. Poisson Distribution | 泊松分布
The Poisson distribution is used to model the number of events occurring within a fixed interval of time, space, or volume. It is characterized by the average rate of occurrence, denoted by λ. The distribution is denoted as X ~ Po(λ). This model is particularly effective for rare events randomly distributed in time or space.
泊松分布用于建模在固定时间、空间或体积间隔内发生的事件次数。它以事件发生的平均速率 λ 为特征，记作 X ~ Po(λ)。该模型在处理随机分布于时间或空间中的稀有事件时尤为有效。
A defining property of the Poisson distribution is that its mean and variance are equal. This equal-dispersion property is a quick check for whether data might follow a Poisson process. The events must occur independently and at a constant average rate, which conceptually matches the binomial under certain limiting conditions.
泊松分布的一个显著性质是其均值与方差相等。这种等离散性质可以快速检验数据是否可能遵循泊松过程。事件必须独立发生，并以恒定的平均速率发生，这在某些极限条件下与二项分布具有概念上的联系。
P(X = x) = (e⁻λ · λˣ) / x!, E(X) = λ, Var(X) = λ
5. Normal Distribution | 正态分布
The normal distribution is the most vital continuous distribution in statistics. It is defined by its mean, μ, and its variance, σ², and is denoted as X ~ N(μ, σ²). The graph of this distribution is the classic bell curve, symmetric about the mean, with the spread determined by the standard deviation.
正态分布是统计学中最重要的连续分布。它由均值 μ 和方差 σ² 定义，记作 X ~ N(μ, σ²)。该分布的图形是经典的钟形曲线，关于均值对称，其离散程度由标准差决定。
To calculate probabilities, any normal distribution must be standardized to the standard normal distribution, often denoted as Z, which has a mean of 0 and a variance of 1. This is achieved by subtracting the mean and dividing by the standard deviation (Z = (X – μ)/σ). Then, a GDC or statistical table is used to find probabilities.
为了计算概率，任何正态分布都必须标准化为标准正态分布（通常记为 Z），其均值为 0，方差为 1。这通过减去均值并除以标准差 (Z = (X – μ)/σ) 实现。然后使用 GDC 或统计表来查找概率。
Z = (X – μ) / σ, P(Z < -a) = P(Z > a), P(Z < a) = Φ(a)
6. Properties of Expectation and Variance | 期望与方差的性质
Understanding how linear transformations affect random variables is a core algebraic skill. If a random variable X is transformed into a new variable Y using the linear relationship Y = aX + b, the expected value of Y is simply the same linear transformation applied to the expected value of X. This reflects the fact that expectation is a linear operator.
理解线性变换如何影响随机变量是一项核心代数技能。如果随机变量 X 通过线性关系 Y = aX + b 变换为新变量 Y，则 Y 的期望值就是对 X 的期望值应用相同的线性变换。这反映了期望是一个线性算子这一事实。
The variance, however, behaves differently. Adding a constant shifts the entire distribution but does not affect its spread. Thus, the variance of Y depends only on the scale factor a, specifically on the square of a. This distinction is a frequent source of exam errors.
然而，方差的行为则不同。加上一个常数会平移整个分布，但不会影响其离散程度。因此，Y 的方差仅取决于缩放因子 a，具体来说是 a 的平方。这种区别是考试中常见的错误来源。
E(aX + b)
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