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 ≤ p_i ≤ 1 and ∑_i p_i = 1

  0 ≤ p_i ≤ 1，且∑_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) = k+110 for k = 0,1,2,3, check that the sum equals 1.

例如公式型：若P(X = k) = k+110，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) = ∑_i x_i P(X = x_i)

It is also called the mean μ. Do not confuse it with the most likely value.

它也称为均值μ。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)) = ∑_i g(x_i) P(X = x_i)

This is directly tested in IB papers, often for g(X)=X², g(X)=2X+1, or linear combinations.

IB试卷中常考此式，例如g(X)=X²、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：

Var(X) = E(X²) - [E(X)]²

The standard deviation is σ = √Var(X). Always state the correct units in context.

标准差为σ = √Var(X)。在应用题中记得注明单位。

Using the previous example: E(X²)=0²(0.1)+1²(0.3)+2²(0.4)+3²(0.2)=3.1, so Var(X)=3.1-1.7²=0.21.

沿用上例：E(X²)=0²(0.1)+1²(0.3)+2²(0.4)+3²(0.2)=3.1，因此Var(X)=3.1-1.7²=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

• Var(aX+b)=a²Var(X)

  Var(aX+b)=a²Var(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

• Var(X+Y)=Var(X)+Var(Y) only if independent

  Var(X+Y)=Var(X)+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,…,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,…,n，则P(X=k)=1/n。其均值与方差为：

μ = n+12, σ² = ^n2-1⁄₁₂

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) = binomnk p^k (1-p)^n-k, k=0,1,…,n

• Mean: μ = np

  均值：μ = np

• Variance: σ² = np(1-p)

  方差：σ² = 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(λ) | 泊松分布Po(λ)

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) = ^{λk e-λ}⁄_k!, k = 0,1,2,…

• Mean: μ = λ

  均值：μ = λ

• Variance: σ² = λ

  方差：σ² = λ

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 ≥ 1)=1-P(X=0).

若题目说“至少一个”，使用P(X ≥ 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 ≤ k), P(X ≥ k), or P(a ≤ X ≤ b). For binomial or Poisson, use the cumulative distribution table or GDC.

考试常要求计算P(X ≤ k)、P(X ≥ k)或P(a ≤ X ≤ 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 ≤ X ≤ b) = F(b) - F(a-1)

For P(X ≥ k), use 1 - P(X ≤ k-1). Remember to adjust endpoints carefully.

对于P(X ≥ k)，利用1 - P(X ≤ 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(X²) before computing variance.

  计算方差前先用精确值求E(X²)。

• Do not use Var(aX+b)=aVar(X); the correct factor is a².

  不要写成Var(aX+b)=aVar(X)，正确系数是a²。

• 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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