Random Variables and Their Distributions | 随机变量及其分布

📚 Random Variables and Their Distributions | 随机变量及其分布

In A-level statistics, the concept of a random variable is the bridge between raw data and probability models. Understanding how to define, represent, and manipulate random variables is essential for solving exam questions on probability distributions.

在A-level统计学中，随机变量是连接原始数据与概率模型的桥梁。理解如何定义、表示和处理随机变量，对于解答概率分布相关的考试题目至关重要。

1. What is a Random Variable? | 什么是随机变量？

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 or Y.

随机变量是取值于随机现象数值结果的变量，通常用大写字母如X或Y表示。

For example, when rolling a fair six-sided die, let X be the number shown. The possible values of X are 1, 2, 3, 4, 5, 6. Each value has a probability of 1/6.

例如，掷一颗均匀六面骰子时，令X表示朝上的点数。X的可能取值为1、2、3、4、5、6，每个取值的概率均为1/6。

Random variables map outcomes from a sample space to real numbers.
随机变量将样本空间中的结果映射到实数。
The set of possible values is called the support of the random variable.
所有可能取值的集合称为随机变量的支撑集。

2. Discrete vs Continuous Random Variables | 离散与连续随机变量

A discrete random variable takes a countable set of values, often integers. Examples include the number of heads in 10 coin flips or the number of cars passing a point in an hour.

离散随机变量取值为可数集合，通常是整数。例如抛10次硬币出现正面的次数，或一小时内通过某地的汽车数量。

A continuous random variable takes values in an interval or union of intervals. Examples include height, weight, or waiting time. It can assume infinitely many uncountable values.

连续随机变量取值于一个区间或多个区间的并集。例如身高、体重或等待时间，它可以取无穷多个不可数的值。

For discrete variables, probabilities are assigned to individual values.
对于离散变量，概率分配给每个单独的取值。
For continuous variables, probabilities are assigned to intervals, not individual points.
对于连续变量，概率分配给区间，而非单独的点。

3. Probability Distribution Function for Discrete Variables | 离散变量的概率分布函数

For a discrete random variable X, the probability distribution function (p.d.f.) lists each possible value x and its corresponding probability P(X = x).

对于离散随机变量X，概率分布函数列出每一个可能值x及其对应概率P(X = x)。

The function p(x) must satisfy two conditions:

函数p(x)必须满足两个条件：

0 ≤ p(x) ≤ 1 and Σ p(x) = 1

Example: Let X be the sum of two fair dice. Then P(X = 2) = 1/36, P(X = 3) = 2/36, etc. These probabilities sum to 1.

例如：令X为两颗均匀骰子的点数之和，则P(X = 2) = 1/36，P(X = 3) = 2/36，依此类推，所有概率之和为1。

A probability distribution table is often used to present p(x).
概率分布表常用于展示p(x)。
Graphically, vertical line graphs or bar charts are used.
在图形上，常用垂直线图或条形图表示。

4. Cumulative Distribution Function | 累积分布函数

The cumulative distribution function (c.d.f.) F(x) gives the probability that X takes a value less than or equal to x:

累积分布函数F(x)给出X取值小于或等于x的概率：

F(x) = P(X ≤ x)

For a discrete random variable, F(x) is a step function that increases by jumps at each possible value.

对于离散随机变量，F(x)是阶梯函数，在每个可能取值处发生跳跃。

For a continuous random variable, F(x) is a smooth non-decreasing function, and the probability of an interval [a, b] is given by F(b) – F(a).

对于连续随机变量，F(x)是平滑的非递减函数，区间[a, b]的概率为F(b) – F(a)。

F(x) always lies between 0 and 1.
F(x)始终在0到1之间。
As x → -∞, F(x) → 0; as x → +∞, F(x) → 1.
当x → -∞时，F(x) → 0；当x → +∞时，F(x) → 1。

5. Expected Value and Variance | 期望与方差

The expected value (or mean) of a discrete random variable X is a weighted average of its possible values, weighted by their probabilities.

离散随机变量X的期望值（或均值）是其所有可能取值按各自概率加权的平均值。

E(X) = Σ x · P(X = x)

The variance measures the spread of the distribution around the mean.

方差度量分布围绕均值的离散程度。

Var(X) = E[(X – μ)²] = Σ (x – μ)² · P(X = x)

Equivalently, Var(X) = E(X²) – [E(X)]². This form is often easier to compute from a table.

等价地，Var(X) = E(X²) – [E(X)]²。在表格计算中此形式通常更简便。

The standard deviation is √Var(X).
标准差为√Var(X)。
Expected value is also called the first moment; E(X²) is the second moment.
期望值也称为一阶矩；E(X²)是二阶矩。

6. Properties of Expectation and Variance | 期望与方差的性质

For constants a and b, the following linearity properties hold for any random variable X:

对于常数a和b，以下线性性质对任意随机变量X均成立：

E(aX + b) = aE(X) + b

Var(aX + b) = a²Var(X)

Note that adding a constant b does not affect the variance, because it shifts the distribution but does not change its spread.

注意：加上常数b不影响方差，因为平移分布不改变离散程度。

For two independent random variables X and Y, E(X + Y) = E(X) + E(Y).
若X和Y相互独立，则E(X + Y) = E(X) + E(Y)。
Also, Var(X + Y) = Var(X) + Var(Y) when independent.
同样，独立时Var(X + Y) = Var(X) + Var(Y)。

7. Binomial Distribution | 二项分布

A binomial distribution models the number of successes in a fixed number n of independent trials, each with the same success probability p.

二项分布用于建模在固定次数n次独立试验中成功的次数，每次试验的成功概率p相同。

X ~ B(n, p)

Its probability function is:

其概率函数为：

P(X = r) = C(n, r) pʳ (1 – p)ⁿ⁻ʳ, r = 0, 1, …, n

The mean and variance are:

均值和方差为：

E(X) = np, Var(X) = np(1 – p)

Conditions: fixed n, independent trials, two outcomes, constant p.
使用条件：固定n、独立试验、两种结果、p恒定。
The binomial approximation is valid only when these conditions are met.
仅在满足这些条件时二项近似才有效。

8. Poisson Distribution | 泊松分布

A Poisson distribution models the number of events occurring in a fixed interval of time or space, given that events occur at a constant average rate λ and independently.

泊松分布用于建模在固定时间或空间区间内发生的事件次数，要求事件以恒定平均速率λ发生且相互独立。

X ~ Poisson(λ)

Its probability function is:

其概率函数为：

P(X = r) = e⁻λ λʳ / r!, r = 0, 1, 2, …

The mean and variance are both equal to λ:

均值和方差都等于λ：

E(X) = λ, Var(X) = λ

Useful for rare events: number of accidents, calls in a call centre, particles in a solution.
适用于稀有事件：事故次数、呼叫中心来电数、溶液中的粒子数。
If X ~ Poisson(λ₁) and Y ~ Poisson(λ₂) are independent, then X + Y ~ Poisson(λ₁ + λ₂).
若X ~ Poisson(λ₁)和Y ~ Poisson(λ₂)独立，则X + Y ~ Poisson(λ₁ + λ₂)。

9. Normal Distribution | 正态分布

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

正态分布是最重要的连续分布。它对称、呈钟形，完全由均值μ和标准差σ描述。

X ~ N(μ, σ²)

Its probability density function has the form:

其概率密度函数形式为：

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

The curve is symmetric about x = μ.
曲线关于x = μ对称。
The total area under the curve is 1.
曲线下总面积为1。
Approximately 68% of data lies within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
约68%的数据落在μ ± σ内，95%落在μ ± 2σ内，99.7%落在μ ± 3σ内。

10. Standardisation of Normal Distribution | 正态分布标准化

To compute probabilities for any normal distribution, we convert X to the standard normal variable Z:

为计算任意正态分布的概率，我们将X转换为标准正态变量Z：

Z = (X – μ) / σ

Then Z ~ N(0, 1), the standard normal distribution with mean 0 and variance 1.

于是Z ~ N(0, 1)，即均值为0、方差为1的标准正态分布。

For example, if X ~ N(50, 100), then P(X < 60) = P(Z < (60 - 50)/10) = P(Z < 1).

例如，若X ~ N(50, 100)，则P(X < 60) = P(Z < (60 - 50)/10) = P(Z < 1)。

Always draw a diagram of the normal curve and shade the area required.
画正态曲线图并标出所需区域总是有帮助的。
Use symmetry: P(Z < -a) = 1 - P(Z < a).
利用对称性：P(Z < -a) = 1 - P(Z < a)。

11. Using Distribution Tables | 使用分布表

Exam questions often give standard normal tables that provide P(Z < z) for non-negative z values.

考试题目常提供标准正态分布表，给出非负z值的P(Z < z)。

To find P(Z > z), use the complement rule: P(Z > z) = 1 – P(Z < z).

要求P(Z > z)，使用互补法则：P(Z > z) = 1 – P(Z < z)。

For negative z, use symmetry: P(Z < -z) = P(Z > z).

对于负z，利用对称性：P(Z < -z) = P(Z > z)。

Event	Calculation
P(Z < 1.25)	Read table directly = 0.8944
P(Z > 1.25)	1 – 0.8944 = 0.1056
P(Z < -1.25)	P(Z > 1.25) = 0.1056
P(-1 < Z < 1)	0.8413 – 0.1587 = 0.6826

Always check whether the table gives area to the left or to the right.
始终检查表格给出的是左侧面积还是右侧面积。
Interpolation may be required if your z-value is not exactly in the table.
若z值未精确出现在表中，可能需要插值。

12. Selecting the Correct Distribution | 选择正确的分布

Choosing the right distribution is a key exam skill. Read the problem carefully and identify the underlying process.

选择正确的分布是重要的考试技能。仔细阅读题目并识别潜在过程。

Fixed number of independent trials with two outcomes → Binomial.
固定次数的独立试验且只有两种结果 → 二项分布。
Events occurring randomly at a constant average rate → Poisson.
事件以恒定平均速率随机发生 → 泊松分布。
Symmetric continuous measurements, often with known mean and variance → Normal.
对称的连续测量值，通常已知均值和方差 → 正态分布。
If n is large and p is small, Poisson can approximate Binomial.
若n大且p小，泊松分布可近似二项分布。
If np and n(1-p) are both at least 5, Normal can approximate Binomial.
若np和n(1-p)均不小于5，正态分布可近似二项分布。

Always state the distribution you are using in your solution, and show how you identify the parameters.

在解答中始终说明你使用的分布，并展示如何确定参数。

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Random Variables and Their Distributions | 随机变量及其分布