📚 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
bability 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。
, 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。
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
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
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
s 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
nd 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。
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
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。
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