Probability Distributions | 概率分布

📚 Probability Distributions | 概率分布

A probability distribution describes how the total probability of 1 is spread across the possible values of a random variable. In A-Level Mathematics, you will encounter both discrete and continuous distributions, learn how to calculate probabilities, expected values, and variances, and understand how to model real-world situations. This article covers the key distributions in the Edexcel syllabus: the discrete uniform, binomial, and Poisson distributions, along with the continuous uniform and normal distributions. You will also learn about approximations and how to choose the correct model for a given scenario.

概率分布描述了总概率 1 如何分配给随机变量的各个可能取值。在 A-Level 数学中，你会学习离散和连续分布，学会计算概率、期望值和方差，并理解如何对现实情境建模。本文涵盖 Edexcel 考纲中的关键分布：离散均匀分布、二项分布、泊松分布，以及连续均匀分布和正态分布。你还会学到近似方法以及如何为给定情境选择正确的模型。

1. Discrete Random Variables | 离散随机变量

A discrete random variable (DRV) takes a countable number of distinct values. These values are usually integers, and each has an associated probability. The sum of all probabilities in the distribution must equal 1. For a DRV X, we denote P(X = x) as the probability that X takes the value x. The support is the set of values where P(X = x) > 0.

离散随机变量（DRV）取有限个或可数个互不相同的值。这些值通常是整数，每个值都有一个对应的概率。分布中所有概率之和必须等于 1。对于离散随机变量 X，我们用 P(X = x) 表示 X 取值为 x 的概率。支撑集是所有使 P(X = x) > 0 的 x 的集合。

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2. Probability Mass Function | 概率质量函数

The probability mass function (PMF) specifies P(X = x) for each possible value x. A valid PMF satisfies two conditions: 0 ≤ P(X = x) ≤ 1 for all x, and Σ P(X = x) = 1 over all possible values. The PMF can be displayed in a table or as a function. In a table, the first row lists the values x, and the second row lists the corresponding probabilities P(X = x).

概率质量函数（PMF）为每个可能的取值 x 指定了 P(X = x)。一个有效的 PMF 满足两个条件：对所有 x 有 0 ≤ P(X = x) ≤ 1，并且所有可能取值的 Σ P(X = x) = 1。PMF 可以用表格或函数形式展示。在表格中，第一行列出了取值 x，第二行列出了相应的概率 P(X = x)。

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3. Cumulative Distribution Function | 累积分布函数

The cumulative distribution function (CDF) gives the probability that the random variable is less than or equal to a certain value: F(x) = P(X ≤ x). For a discrete random variable, F(x) is a step function that increases at each value in the support. It is often used to find probabilities of intervals, e.g., P(a < X ≤ b) = F(b) − F(a). The CDF is always non‑decreasing and ends at 1.

累积分布函数（CDF）给出随机变量小于或等于某个值的概率：F(x) = P(X ≤ x)。对于离散随机变量，F(x) 是一个阶梯函数，在支撑集的每个值处跳跃上升。它常用来求区间的概率，例如 P(a < X ≤ b) = F(b) − F(a)。CDF 总是非降的，最终达到 1。

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4. Expectation and Variance | 期望与方差

The expectation (or expected value) of a discrete random variable X is a measure of its central tendency, calculated as E(X) = Σ x · P(X = x). The variance measures the spread: Var(X) = E(X²) − [E(X)]², where E(X²) = Σ x² · P(X = x). The standard deviation is σ = √Var(X). For a linear transformation Y = aX + b, we have E(Y) = aE(X) + b and Var(Y) = a² Var(X).

离散随机变量 X 的期望（或期望值）是衡量其集中趋势的指标，计算公式为 E(X) = Σ x · P(X = x)。方差衡量离散程度：Var(X) = E(X²) − [E(X)]²，其中 E(X²) = Σ x² · P(X = x)。标准差为 σ = √Var(X)。对于线性变换 Y = aX + b，有 E(Y) = aE(X) + b 且 Var(Y) = a² Var(X)。

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5. Uniform Distribution (Discrete) | 均匀分布（离散）

A discrete uniform distribution occurs when all outcomes are equally likely. If X can take n distinct values, then P(X = x) = 1/n for each value. For example, rolling a fair six‑sided die gives each outcome probability 1/6. The expectation is the average of the smallest and largest values, E(X) = (a + b)/2, where the values range from a to b. The variance is Var(X) = (n² − 1)/12 where n = b − a + 1.

离散均匀分布是所有结果等可能发生时的分布。如果 X 可以取 n 个不同的值，那么每个值的概率为 P(X = x) = 1/n。例如，掷一个均匀六面骰子，每个结果的概率都是 1/6。期望值是最小值和最大值的平均值，E(X) = (a + b)/2，其中取值范围从 a 到 b。方差为 Var(X) = (n² − 1)/12，其中 n = b − a + 1。

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6. Binomial Distribution | 二项分布

A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p. We write X ~ B(n, p). The PMF is P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, for r = 0, 1, 2, …, n. The binomial coefficient ⁿCᵣ is the number of ways to choose r successes from n trials. Key conditions: fixed number of trials, independence, constant probability of success, and each trial has only two outcomes. The expectation is E(X) = np, and variance Var(X) = np(1 − p).

二项分布用于描述在固定次数独立试验中成功的次数，每次试验成功的概率 p 相同。记为 X ~ B(n, p)。概率质量函数为 P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，其中 r = 0, 1, 2, …, n。二项系数 ⁿCᵣ 是从 n 次试验中选出 r 次成功的方法数。关键条件：试验次数固定、试验间独立、成功概率恒定、每次试验只有两种结果。期望为 E(X) = np，方差为 Var(X) = np(1 − p)。

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7. Poisson Distribution | 泊松分布

The Poisson distribution models the number of events occurring in a fixed interval of time or space, assuming events happen independently and at a constant average rate. We write X ~ Po(λ), where λ is the mean number of occurrences. The PMF is P(X = r) = (e⁻ᵯ · λʳ) / r!, for r = 0, 1, 2, … The mean and variance are both λ: E(X) = Var(X) = λ. The Poisson distribution is often used for rare events, such as the number of calls to a call centre per minute.

泊松分布用于描述在固定的时间或空间间隔内发生的事件次数，假设事件独立发生且以恒定的平均速率出现。记为 X ~ Po(λ)，其中 λ 是事件发生的平均次数。概率质量函数为 P(X = r) = (e⁻ᵯ · λʳ) / r!，r = 0, 1, 2, …。均值和方差都是 λ：E(X) = Var(X) = λ。泊松分布常用于稀有事件，例如每分钟呼叫中心的来电数量。

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8. Continuous Random Variables | 连续随机变量

A continuous random variable can take any value within an interval. Probabilities are found using a probability density function (PDF), f(x). The total area under the PDF curve is 1. The probability of the variable falling in an interval [a, b] is the definite integral ∫ₐᵇ f(x) dx. Unlike discrete variables, P(X = c) = 0 for any single value c. The cumulative distribution function is F(x) = P(X ≤ x) = ∫_{-∞}ˣ f(t) dt.

连续随机变量可以在某个区间内取任意值。概率通过概率密度函数（PDF）f(x) 求得。PDF 曲线下的总面积为 1。变量落在区间 [a, b] 内的概率是定积分 ∫ₐᵇ f(x) dx。与离散变量不同，对于任何单个值 c，P(X = c) = 0。累积分布函数为 F(x) = P(X ≤ x) = ∫_{-∞}ˣ f(t) dt。

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9. Normal Distribution | 正态分布

The normal distribution is a symmetric continuous distribution defined by its mean μ and variance σ². Its PDF is (1/(σ√(2π))) e^(-(x − μ)²/(2σ²)). The curve is bell‑shaped and centred at μ. About 68% of data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. Many natural phenomena, such as heights and measurement errors, are approximately normally distributed.

正态分布是一种由均值 μ 和方差 σ² 确定的对称连续分布。其 PDF 为 (1/(σ√(2π))) e^(-(x − μ)²/(2σ²))。曲线呈钟形，以 μ 为中心。大约 68% 的数据落在均值 ±1σ 内，95% 落在 ±2σ 内，99.7% 落在 ±3σ 内。许多自然现象，如身高和测量误差，近似服从正态分布。

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10. Standard Normal Distribution | 标准正态分布

The standard normal distribution has mean 0 and variance 1, denoted Z ~ N(0, 1). Any normal variable X ~ N(μ, σ²) can be standardised using Z = (X − μ)/σ. This allows us to use standard normal tables to find probabilities. The CDF of Z is often denoted Φ(z). To find P(X < a), calculate z = (a − μ)/σ, then look up Φ(z). For intervals, P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ).

标准正态分布均值为 0、方差为 1，记作 Z ~ N(0, 1)。任何正态变量 X ~ N(μ, σ²) 都可以通过 Z = (X − μ)/σ 标准化。这使得我们可以用标准正态分布表查找概率。Z 的 CDF 常记为 Φ(z)。要计算 P(X < a)，先计算 z = (a − μ)/σ，然后查表得 Φ(z)。对于区间，P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ)。

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11. Approximations | 近似方法

Under certain conditions, one distribution can approximate another. A binomial distribution B(n, p) can be approximated by a Poisson distribution Po(λ = np) if n is large and p is small (typically n > 50 and np < 5). A binomial distribution can also be approximated by a normal distribution N(np, np(1 − p)) if np > 5 and n(1 − p) > 5. A Poisson distribution Po(λ) can be approximated by N(λ, λ) when λ is large (usually λ > 10). Continuity corrections are required when approximating a discrete distribution with a continuous one.

在特定条件下，一种分布可以近似另一种分布。如果 n 很大且 p 很小（通常 n > 50 且 np < 5），二项分布 B(n, p) 可用泊松分布 Po(λ = np) 近似。如果 np > 5 且 n(1 − p) > 5，二项分布也可用正态分布 N(np, np(1 − p)) 近似。当 λ 较大（通常 λ > 10）时，泊松分布 Po(λ) 可用 N(λ, λ) 近似。用连续分布近似离散分布时，需要进行连续性校正。

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12. Choosing the Right Distribution | 选择合适的分布

To decide which distribution to use, examine the context. If there are a fixed number of independent trials with two outcomes, use binomial. If events occur randomly at a constant average rate in time or space, use Poisson. If data are continuous and symmetric around a mean with no bounds, a normal distribution may be appropriate. For equally likely outcomes from a finite set, use discrete uniform. Always check that the modelling conditions are satisfied before applying a distribution.

要决定使用哪种分布，需分析具体情境。如果有固定次数的独立试验且每次只有两种结果，则使用二项分布。如果事件在时间或空间上以恒定平均速率随机发生，则使用泊松分布。如果数据连续且关于均值对称且无界限，正态分布可能合适。对于有限集合中等可能的结果，使用离散均匀分布。在应用分布之前，务必检查建模条件是否满足。

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