Probability Distributions: Core Concepts | 概率分布的基本概念归纳

📚 Probability Distributions: Core Concepts | 概率分布的基本概念归纳

A probability distribution is a mathematical description that assigns probabilities to the possible outcomes of a random experiment. It tells us not only what can happen, but how likely each outcome is.

概率分布是对随机试验中各种可能结果赋予概率的数学描述。它不仅告诉我们可能发生什么，还告诉我们每个结果发生的可能性有多大。

In A-level mathematics, mastering the core concepts of probability distributions is essential for solving problems in statistics, hypothesis testing, and modelling real-world situations. This article groups the fundamental ideas into a clear revision guide.

在 A-level 数学中，掌握概率分布的基本概念对于解决统计、假设检验以及模拟现实情境的问题至关重要。本文将核心思想整理成一份清晰的复习指南。

1. Random Variables | 随机变量

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. It is usually denoted by a capital letter such as X, and its observed value by a lowercase letter such as x.

随机变量是其可能取值为随机现象数值结果的变量。通常用大写字母如 X 表示，其观测值用小写字母如 x 表示。

For example, if we toss two coins and count the number of heads, the random variable X could take values 0, 1, or 2. The value of X is not known until the experiment is performed.

例如，如果我们抛两枚硬币并数正面的数量，随机变量 X 可能取值为 0、1 或 2。在实验完成之前，X 的值是未知的。

Random variables are classified into two main types: discrete and continuous. This classification determines how we describe their probability distributions.

随机变量主要分为两大类：离散型和连续型。这种分类决定了我们如何描述它们的概率分布。

2. Discrete vs Continuous | 离散与连续

A discrete random variable can take only a countable number of distinct values, often integers. Examples include the number of students in a class, the number of emails received in a day, or the number of defective items in a batch.

离散随机变量只能取有限个或可数个不同的值，通常是整数。例如班级学生的数量、一天收到的电子邮件数量，或一批产品中的次品数量。

A continuous random variable can take any value within an interval or range. Examples include height, weight, time, or temperature. For a continuous variable, the probability of any single exact value is theoretically zero.

连续随机变量可以在某个区间或范围内取任意值。例如身高、体重、时间或温度。对于连续变量，任何单个精确值的概率理论上为零。

The difference is important because the tools used to calculate probabilities differ: sums for discrete variables and integrals for continuous variables.

这一差异非常重要，因为计算概率的工具不同：离散型使用求和，连续型使用积分。

3. Probability Mass Function (PMF) | 概率质量函数

The probability mass function, or PMF, is used for discrete random variables. It gives the probability that the random variable X equals a specific value x, written as p(x) = P(X = x).

概率质量函数（PMF）用于离散随机变量。它给出随机变量 X 等于某个特定值 x 的概率，记为 p(x) = P(X = x)。

A valid PMF must satisfy two conditions: first, every probability is between 0 and 1 inclusive; second, the sum of all probabilities over all possible values equals 1.

一个有效的 PMF 必须满足两个条件：第一，每个概率都在 0 到 1 之间（含端点）；第二，所有可能取值的概率之和等于 1。

Condition 1: 0 ≤ p(x) ≤ 1 for every x.

条件 1：对每一个 x，有 0 ≤ p(x) ≤ 1。
Condition 2: Σ p(x) = 1 over all x.

条件 2：对所有 x 求和，Σ p(x) = 1。

For example, if X represents the number of heads when two coins are tossed, then P(X = 0) = 0.25, P(X = 1) = 0.5, and P(X = 2) = 0.25. These probabilities add to 1.

例如，若 X 表示抛两枚硬币后正面的数量，则 P(X = 0) = 0.25，P(X = 1) = 0.5，P(X = 2) = 0.25。这些概率之和等于 1。

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

The probability density function, or PDF, describes a continuous random variable. It is a function f(x) that is always non-negative, and the total area under its curve is exactly 1.

概率密度函数（PDF）用于描述连续随机变量。它是一个总是非负的函数 f(x)，其曲线下的总面积恰好等于 1。

Probability is interpreted as the area under the curve over an interval. Thus P(a ≤ X ≤ b) is the area from x = a to x = b under f(x).

概率被解释为曲线下某个区间内的面积。因此 P(a ≤ X ≤ b) 是 f(x) 曲线下从 x = a 到 x = b 的面积。

P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx

Because the probability of any single point is zero, inequalities such as X < a and X ≤ a give the same probability for a continuous variable.

由于任意单点的概率为零，对于连续变量，X < a 和 X ≤ a 所给出的概率是相同的。

The PDF itself can be greater than 1 in some regions, as long as the total area is 1. The value of f(x) is a density, not a probability.

PDF 本身在某些区域可以大于 1，只要总面积为 1 即可。f(x) 的值是密度，而不是概率。

5. 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 a specified value x. It is defined as F(x) = P(X ≤ x).

累积分布函数记为 F(x)，它给出随机变量 X 小于或等于某个指定值 x 的概率，其定义为 F(x) = P(X ≤ x)。

For a discrete random variable, the CDF is obtained by summing the PMF up to x:

对于离散随机变量，CDF 通过对 PMF 累加至 x 得到：

F(x) = Σₖ₌₋∞ˣ p(k)

For a continuous random variable, the CDF is the integral of the PDF from negative infinity to x:

对于连续随机变量，CDF 是 PDF 从负无穷到 x 的积分：

F(x) = ∫₋∞ˣ f(t) dt

The CDF has three key properties: it is non-decreasing, it approaches 0 as x → −∞, and it approaches 1 as x → +∞. This makes it a convenient tool for calculating probabilities of intervals.

CDF 有三个关键性质：它是不减函数，当 x → −∞ 时趋近于 0，当 x → +∞ 时趋近于 1。这使它成为计算区间概率的便捷工具。

P(a < X ≤ b) = F(b) − F(a)

6. Expected Value | 期望值

The expected value, denoted E(X) or μ, is a measure of the centre of a probability distribution. It represents the long-run average value of the random variable over many repetitions of the experiment.

期望值记为 E(X) 或 μ，是衡量概率分布中心位置的指标。它表示在多次重复试验中随机变量的长期平均值。

For a discrete random variable, the expected value is the weighted sum of all possible values, where the weights are their probabilities:

对于离散随机变量，期望值是所有可能取值的加权和，权重为对应概率：

E(X) = Σ x · p(x)

For a continuous random variable, the expected value is calculated using the integral:

对于连续随机变量，期望值通过积分计算：

E(X) = ∫₋∞∞ x · f(x) dx

The expected value is linear. For any constants a and b, E(aX + b) = aE(X) + b. This property is extremely useful in algebraic manipulations.

期望值具有线性性质。对于任意常数 a 和 b，有 E(aX + b) = aE(X) + b。这一性质在代数运算中非常有用。

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

Variance measures the spread or dispersion of a probability distribution around its mean. It is defined as the expected value of the squared deviation from the mean:

方差衡量概率分布在其均值周围的离散程度。它定义为离均差平方的期望值：

Var(X) = E[(X − μ)²]

An equivalent and often easier formula is:

一个等价且通常更容易计算的公式是：

Var(X) = E(X²) − [E(X)]²

The standard deviation, denoted σ, is the square root of the variance. It has the same units as the original data, which makes it easier to interpret.

标准差记为 σ，是方差的平方根。它与原始数据具有相同的单位，因此更容易解释。

For variance, the scaling rule is Var(aX + b) = a²Var(X). Note that adding a constant b does not change the variance, because it shifts the distribution without changing its spread.

方差的伸缩规则为 Var(aX + b) = a²Var(X)。注意加上常数 b 不会改变方差，因为平移分布不会改变其离散程度。

8. Common Discrete Distributions: Binomial and Poisson | 常见离散分布：二项与泊松

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. If X ~ B(n, p), then:

二项分布用于模拟在固定次数的独立试验中成功的次数，每次试验的成功概率相同。若 X ~ B(n, p)，则：

P(X = x) = C(n, x) pˣ (1 − p)ⁿ⁻ˣ

Its mean and variance are:

其均值和方差为：

E(X) = np, Var(X) = np(1 − p)

The Poisson distribution models the number of rare events occurring in a fixed interval of time or space. If X ~ Po(λ), then:

泊松分布用于模拟在固定时间或空间区间内稀有事件发生的次数。若 X ~ Po(λ)，则：

P(X = x) = e⁻λ λˣ / x!

For a Poisson distribution, the mean and variance are both equal to λ:

对于泊松分布，均值和方差都等于 λ：

E(X) = λ, Var(X) = λ

The Poisson distribution is also used as an approximation to the binomial when n is large and p is small, with λ = np.

当 n 很大而 p 很小时，泊松分布也可用作二项分布的近似，此时 λ = np。

9. Common Continuous Distribution: Normal | 常见连续分布：正态

The normal distribution is the most important continuous distribution in statistics. It is symmetric, bell-shaped, and fully described by its mean μ and standard deviation σ.

正态分布是统计学中最重要的连续分布。它对称、钟形，且完全由其均值 μ 和标准差 σ 描述。

The probability density function of a normal distribution is:

正态分布的概率密度函数为：

f(x) = 1 / (σ√(2π)) · e^(−(x − μ)² / (2σ²))

If X ~ N(μ, σ²), we can standardise it by subtracting the mean and dividing by the standard deviation:

若 X ~ N(μ, σ²)，我们可以通过减去均值并除以标准差来将其标准化：

Z = (X − μ) / σ

The standard normal distribution Z ~ N(0, 1) has mean 0 and standard deviation 1. Probabilities are found using standard normal tables or calculators.

标准正态分布 Z ~ N(0, 1) 的均值为 0，标准差为 1。概率通过标准正态分布表或计算器求得。

The empirical rule states that for any normal distribution, approximately 68% of values lie within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.

经验法则指出，对于任何正态分布，大约 68% 的数值位于均值附近 1 个标准差之内，95% 位于 2 个标准差之内，99.7% 位于 3 个标准差之内。

10. Choosing the Right Distribution | 选择正确的分布

Selecting the appropriate probability distribution requires careful thought about the nature of the random variable and the conditions of the experiment.

选择适当的概率分布需要仔细思考随机变量的性质以及实验的条件。

Use the binomial distribution if there are a fixed number n of independent trials, each with two possible outcomes and a constant probability p of success.

如果存在固定次数 n 的独立试验，每次试验只有两种可能结果且成功概率 p 恒定，则使用二项分布。

Use the Poisson distribution for counting the number of events in a fixed interval when events occur independently and at a constant average rate λ.

当事件独立发生且以恒定平均速率 λ 在固定区间内出现时，计数事件次数可使用泊松分布。

Use the normal distribution for continuous measurements such as heights, weights, or errors, especially when the data are roughly symmetric and unimodal.

对于身高、体重或误差等连续测量值，尤其当数据大致对称且单峰时，使用正态分布。

Distribution	Type	Parameters	Mean	Variance
Binomial	Discrete	n, p	np	np(1 − p)
Poisson	Discrete	λ	λ	λ
Normal	Continuous	μ, σ	μ	σ²

Always check the conditions before using a distribution: independence of trials, constant probability, and whether the variable is discrete or continuous. A wrong distribution leads to wrong probabilities and incorrect conclusions.

在使用分布之前一定要检查条件：试验是否独立、概率是否恒定，以及变量是离散型还是连续型。错误的分布会导致错误的概率和不正确的结论。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Probability Distributions: Core Concepts | 概率分布的基本概念归纳