IB Mathematics: Discrete Random Variables Exam Guide | IB数学：离散随机变量考点解析

📚 IB Mathematics: Discrete Random Variables Exam Guide | IB数学：离散随机变量考点解析

A discrete random variable takes countable values, each with an associated probability. In IB Mathematics, understanding its distribution, expectation, variance, and common models is essential for exam success.

离散随机变量取值为可数集合，每个取值伴随一个概率。在IB数学中，掌握其分布、期望、方差以及常见模型是考试得分的核心。

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1. Definition and Probability Distribution | 定义与概率分布

A discrete random variable \(X\) is a variable whose possible values are finite or countably infinite. The probability distribution of \(X\) assigns a probability \(P(X = x)\) to each value \(x\). For a valid distribution, all probabilities must be non-negative and sum to 1.

离散随机变量\(X\)是一个取值有限或可数无穷的变量。\(X\)的概率分布为每个取值\(x\)赋予概率\(P(X = x)\)。一个有效分布必须满足：所有概率非负，且总和为1。

• \(p_i = P(X = x_i)\) for each \(i\)

  \(p_i = P(X = x_i)\)，对每个\(i\)成立

• \(0 \le p_i \le 1\) and \(\sum_i p_i = 1\)

  \(0 \le p_i \le 1\)，且\(\sum_i p_i = 1\)

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2. Representing Distributions: Tables, Graphs, Formulas | 分布表示：表格、图形与公式

In IB exams, discrete distributions are often given as probability tables, bar charts, or algebraic formulas. You must be able to convert between these representations.

IB考试中，离散分布常以概率表、柱状图或代数公式呈现。你必须能够在这些表示形式之间转换。

x	0	1	2	3
P(X=x)	0.1	0.3	0.4	0.2

For a formula example, if \(P(X = k) = \frac{k+1}{10}\) for \(k = 0,1,2,3\), check that the sum equals 1.

例如公式型：若\(P(X = k) = \frac{k+1}{10}\)，其中\(k = 0,1,2,3\)，请验证总和为1。

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3. Expected Value: \(E(X)\) | 期望值\(E(X)\)

The expected value measures the long-run average of the random variable. For a discrete distribution:

期望值衡量随机变量的长期平均水平。对于离散分布：

\(E(X) = \sum_i x_i P(X = x_i)\)

It is also called the mean \(\mu\). Do not confuse it with the most likely value.

它也称为均值\(\mu\)。不要将其与最可能取值混淆。

Example: For the table above, \(E(X)=0(0.1)+1(0.3)+2(0.4)+3(0.2)=1.7\).

例如：对上表，\(E(X)=0(0.1)+1(0.3)+2(0.4)+3(0.2)=1.7\)。

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4. Expected Value of a Function \(E(g(X))\) | 函数期望\(E(g(X))\)

To find the expected value of a transformed variable, apply the function to each value before averaging:

求变换后变量的期望时，先对每个取值套用函数，再求加权平均：

\(E(g(X)) = \sum_i g(x_i) P(X = x_i)\)

This is directly tested in IB papers, often for \(g(X)=X^2\), \(g(X)=2X+1\), or linear combinations.

IB试卷中常考此式，例如\(g(X)=X^2\)、\(g(X)=2X+1\)或线性组合。

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

Variance measures the spread of a distribution around the mean. It can be computed using the shortcut formula:

方差衡量分布相对均值的离散程度。可用简化公式计算：

\(\text{Var}(X) = E(X^2) – [E(X)]^2\)

The standard deviation is \(\sigma = \sqrt{\text{Var}(X)}\). Always state the correct units in context.

标准差为\(\sigma = \sqrt{\text{Var}(X)}\)。在应用题中记得注明单位。

Using the previous example: \(E(X^2)=0^2(0.1)+1^2(0.3)+2^2(0.4)+3^2(0.2)=3.1\), so \(\text{Var}(X)=3.1-1.7^2=0.21\).

沿用上例：\(E(X^2)=0^2(0.1)+1^2(0.3)+2^2(0.4)+3^2(0.2)=3.1\)，因此\(\text{Var}(X)=3.1-1.7^2=0.21\)。

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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 are frequently used:

对于常数\(a\)和\(b\)，以下线性性质经常使用：

• \(E(aX+b)=aE(X)+b\)

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

• \(\text{Var}(aX+b)=a^2\text{Var}(X)\)

  \(\text{Var}(aX+b)=a^2\text{Var}(X)\)

• \(E(X+Y)=E(X)+E(Y)\) for any variables

  \(E(X+Y)=E(X)+E(Y)\)，对任意变量成立

• \(\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)\) only if independent

  \(\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)\)，仅在独立时成立

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7. The Discrete Uniform Distribution | 离散均匀分布

If \(X\) takes values \(1,2,\dots,n\) each with equal probability, then \(P(X=k)=1/n\). The mean and variance are:

若\(X\)等可能地取\(1,2,\dots,n\)，则\(P(X=k)=1/n\)。其均值与方差为：

\(\mu = \frac{n+1}{2}, \quad \sigma^2 = \frac{n^2-1}{12}\)

This model appears in questions involving fair dice, spinners, or random integer selection.

该模型常见于公平骰子、转盘或随机选取整数的问题。

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8. The Binomial Distribution \(B(n,p)\) | 二项分布\(B(n,p)\)

A binomial random variable counts the number of successes in \(n\) independent Bernoulli trials, each with success probability \(p\). Its probability mass function is:

二项随机变量衡量\(n\)次独立伯努利试验中的成功次数，单次成功概率为\(p\)。其概率质量函数为：

\(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k=0,1,\dots,n\)

• Mean: \(\mu = np\)

  均值：\(\mu = np\)

• Variance: \(\sigma^2 = np(1-p)\)

  方差：\(\sigma^2 = np(1-p)\)

Conditions: fixed number of trials, independent trials, two outcomes, constant probability.

条件：试验次数固定、各次独立、只有两种结果、成功概率恒定。

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9. The Poisson Distribution \(Po(\lambda)\) | 泊松分布\(Po(\lambda)\)

The Poisson distribution models the number of rare events in a fixed interval of time or space. Its probability mass function is:

泊松分布用于建模固定时间或空间间隔内稀有事件的发生次数。其概率质量函数为：

\(P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0,1,2,\dots\)

• Mean: \(\mu = \lambda\)

  均值：\(\mu = \lambda\)

• Variance: \(\sigma^2 = \lambda\)

  方差：\(\sigma^2 = \lambda\)

Key property: equal mean and variance. Common examples include arrivals, phone calls, or particles emitted.

关键特征：均值等于方差。常见例子包括到达人数、电话呼叫或粒子发射。

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10. Choosing the Correct Model | 正确选择模型

IB examiners often ask whether a scenario should be modelled by a binomial or Poisson distribution. Use these clues:

IB考官常要求判断一个情境应使用二项分布还是泊松分布。可参考以下提示：

Binomial 二项	Poisson 泊松
Fixed number of trials 试验次数固定	Events occur randomly in time/space 事件在时间/空间中随机发生
Two outcomes per trial 每次试验两种结果	No fixed maximum 没有固定上限
Independent trials 各次独立	Events independent 事件相互独立

If the question says “at least one”, use \(P(X \ge 1)=1-P(X=0)\).

若题目说“至少一个”，使用\(P(X \ge 1)=1-P(X=0)\)。

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11. Cumulative Probabilities and Inequalities | 累积概率与不等式

Exams often require \(P(X \le k)\), \(P(X \ge k)\), or \(P(a \le X \le b)\). For binomial or Poisson, use the cumulative distribution table or GDC.

考试常要求计算\(P(X \le k)\)、\(P(X \ge k)\)或\(P(a \le X \le b)\)。对二项或泊松分布，可使用累积分布表或图形计算器。

\(P(a \le X \le b) = F(b) – F(a-1)\)

For \(P(X \ge k)\), use \(1 – P(X \le k-1)\). Remember to adjust endpoints carefully.

对于\(P(X \ge k)\)，利用\(1 – P(X \le k-1)\)。注意端点调整。

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12. Exam Strategies and Common Errors | 考试策略与常见错误

To maximise marks in this topic, practise careful notation and check conditions before applying a model.

要在该主题上拿到高分，务必练习规范记法，并在套用模型前检查条件。

• Always verify that probabilities sum to 1.

  始终验证概率总和为1。

• Use exact values for \(E(X^2)\) before computing variance.

  计算方差前先用精确值求\(E(X^2)\)。

• Do not use \(\text{Var}(aX+b)=a\text{Var}(X)\); the correct factor is \(a^2\).

  不要写成\(\text{Var}(aX+b)=a\text{Var}(X)\)，正确系数是\(a^2\)。

• For binomial questions, clearly state \(n\) and \(p\).

  二项分布问题中，明确写出\(n\)和\(p\)。

• For Poisson questions, confirm that mean equals variance.

  泊松分布问题中，确认均值等于方差。

By mastering these definitions and techniques, you will be well prepared for IB discrete random variable questions.

熟练掌握以上定义与技巧，你将能在IB离散随机变量考题中从容应对。

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