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^2text{Var}(X))

  (text{Var}(aX+b)=a^2text{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)=atext{Var}(X)); the correct factor is (a^2).

  不要写成(text{Var}(aX+b)=atext{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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