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。

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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。

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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。

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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。

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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。

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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。

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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)。

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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。

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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)。

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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。

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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)。

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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。

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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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Published by TutorHao | Mathematics Revision Series | aleveler.com

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