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。

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)

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。

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)。

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。

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)。

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

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。

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。

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。

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))。

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。

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