Discrete Random Variables | 离散随机变量专题精讲

📚 Discrete Random Variables | 离散随机变量专题精讲

In IB Mathematics, discrete random variables form a core topic in probability and statistics. Mastering this topic requires a clear understanding of probability distributions, expected value, variance, and their applications in real-world contexts.

在IB数学中，离散随机变量是概率与统计部分的核心专题。掌握该专题需要清晰理解概率分布、期望值、方差及其在现实情境中的应用。

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1. What Is a Discrete Random Variable? | 什么是离散随机变量？

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. A discrete random variable takes a countable number of distinct values, often integers, such as the number of heads when tossing three coins.

随机变量是取值取决于随机现象数值结果的变量。离散随机变量取有限个或可数无穷多个互不相同的值，通常为整数，例如抛三枚硬币时出现正面的次数。

Example: Let X be the number of sixes obtained when rolling a fair die twice. The possible values of X are 0, 1, 2. Since these are countable, X is discrete.

例如：设X为掷两次公平骰子得到6点的次数。X的可能取值为0、1、2。由于这些值可数，因此X是离散的。

X ∈ {0, 1, 2, … , n} 或 X ∈ {x₁, x₂, x₃, …}

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2. Probability Distribution Function P(X = x) | 概率分布函数 P(X = x)

For a discrete random variable X, the probability distribution is a list of all possible values x and their associated probabilities P(X = x). This is also called the probability mass function (PMF).

对于离散随机变量X，其概率分布列出所有可能的取值x及对应概率P(X=x)，也称为概率质量函数（PMF）。

A valid PMF must satisfy two conditions:

有效的PMF必须满足两个条件：

• 0 ≤ P(X = x) ≤ 1 for every possible value x.

  对于每一个可能取值x，有0 ≤ P(X = x) ≤ 1。

• ∑ P(X = x) = 1 over all possible values of x.

  对所有可能取值x求和，∑ P(X = x) = 1。

Example: The PMF of a fair coin tossed twice, where X counts the number of heads, is:

例如：抛两次公平硬币，X表示正面次数，其PMF为：

x	0	1	2
P(X = x)	1/4	1/2	1/4

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3. Cumulative Distribution Function | 累积分布函数

The cumulative distribution function (CDF) for a discrete random variable is defined as F(x) = P(X ≤ x). It gives the probability that the variable takes a value less than or equal to x.

离散随机变量的累积分布函数（CDF）定义为F(x) = P(X ≤ x)，表示变量取值小于或等于x的概率。

For the coin-toss example, the CDF is:

对于抛硬币的例子，CDF为：

x	x < 0	0 ≤ x < 1	1 ≤ x < 2	x ≥ 2
F(x)	0	1/4	3/4	1

Key property: F(b) − F(a−) = P(a ≤ X ≤ b).

关键性质：F(b) − F(a−) = P(a ≤ X ≤ b)。

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4. Expected Value E(X) | 期望值 E(X)

The expected value (or mean) of a discrete random variable X is the weighted average of all possible values, where the weights are the probabilities. It is denoted by E(X) or μ.

离散随机变量X的期望值（或均值）是所有可能取值的加权平均，权重为对应概率，记作E(X)或μ。

E(X) = ∑ x · P(X = x)

Interpretation: E(X) represents the long-run average value of the random variable over many repetitions of the experiment.

含义：E(X)表示在大量重复试验中随机变量的长期平均值。

Example: For the number of heads in two coin tosses,

例如：两次抛硬币出现正面次数，

E(X) = 0 × (1/4) + 1 × (1/2) + 2 × (1/4) = 1

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

The variance measures the spread of the distribution around the mean. It is defined as Var(X) = E[(X − μ)²]. A useful computational formula is:

方差衡量分布围绕均值的离散程度，定义为Var(X) = E[(X − μ)²]。一个常用的计算公式为：

Var(X) = E(X²) − [E(X)]²

where E(X²) = ∑ x² · P(X = x). The standard deviation is σ = √Var(X).

其中E(X²) = ∑ x² · P(X = x)。标准差为σ = √Var(X)。

Using the coin-toss example: E(X²) = 0² × (1/4) + 1² × (1/2) + 2² × (1/4) = 1.5, so Var(X) = 1.5 − 1² = 0.5, and σ = √0.5 ≈ 0.707.

使用抛硬币的例子：E(X²) = 0² × (1/4) + 1² × (1/2) + 2² × (1/4) = 1.5，因此Var(X) = 1.5 − 1² = 0.5，σ = √0.5 ≈ 0.707。

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6. Properties of E(X) and Var(X) | 期望与方差的性质

Let X be a discrete random variable, and let a and b be constants. Then:

设X为离散随机变量，a和b为常数，则：

• E(aX + b) = aE(X) + b

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

• Var(aX + b) = a²Var(X)

  Var(aX + b) = a²Var(X)

• E(a) = a

  E(a) = a

• Var(a) = 0

  Var(a) = 0

These properties are frequently tested in IB exam questions involving linear transformations of variables.

这些性质在IB考试中经常与变量的线性变换结合考查。

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7. Uniform Discrete Distribution | 离散均匀分布

A discrete uniform distribution over values x₁, x₂, …, xₙ assigns equal probability 1/n to each value. Its mean and variance are:

离散均匀分布对取值x₁, x₂, …, xₙ中的每个值赋予相同概率1/n。其均值和方差为：

E(X) = (x₁ + x₂ + … + xₙ) / n

Var(X) = (1/n) ∑ xᵢ² − [E(X)]²

For example, rolling a fair six-sided die gives a uniform distribution over 1, 2, 3, 4, 5, 6, with mean 3.5 and variance 35/12.

例如，掷一颗公平的六面骰子，在1、2、3、4、5、6上服从均匀分布，均值为3.5，方差为35/12。

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8. Binomial Distribution | 二项分布

If a random variable X counts the number of successes in n independent trials, each with probability p of success, then X ~ B(n, p). Its PMF is:

如果随机变量X表示n次独立试验中成功的次数，每次成功概率为p，则X ~ B(n, p)。其PMF为：

P(X = r) = ⁿCᵣ · pʳ · (1 − p)ⁿ⁻ʳ, r = 0, 1, 2, …, n

The mean and variance of a binomial variable are:

二项变量的均值与方差为：

E(X) = np, Var(X) = np(1 − p)

Example: A multiple-choice test has 10 questions, each with 4 options. Guessing every answer gives X ~ B(10, 0.25). The expected number of correct answers is 10 × 0.25 = 2.5.

例如：一份10道题的单选题测试，每题4个选项。随机猜测时X ~ B(10, 0.25)。答对的期望数为10 × 0.25 = 2.5。

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9. Using the Binomial Table or Calculator | 使用二项分布表或计算器

In IB exams, you are expected to compute binomial probabilities efficiently using a GDC (graphical display calculator) or tables. Typical questions involve P(X = r), P(X ≤ r), or P(X > r).

在IB考试中，你需要使用图形计算器（GDC）或表格高效计算二项概率。典型问题涉及P(X = r)、P(X ≤ r)或P(X > r)。

Key relation: P(X > r) = 1 − P(X ≤ r).

关键关系：P(X > r) = 1 − P(X ≤ r)。

Ensure you know your GDC menu: binomialpdf for P(X = r) and binomialcdf for P(X ≤ r). Make sure the parameters n and p are entered correctly.

务必熟悉计算器菜单：binomialpdf用于P(X = r)，binomialcdf用于P(X ≤ r)。确保参数n和p输入正确。

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10. Transforming Variables and Expectation of a Function | 变量变换与函数期望

For a function g(X), the expected value is given by E[g(X)] = ∑ g(x) · P(X = x). This works for any function g, not only linear ones.

对于函数g(X)，其期望为E[g(X)] = ∑ g(x) · P(X = x)。这适用于任意函数g，不只是线性函数。

Important note: In general, E(X²) ≠ [E(X)]², and E(1/X) ≠ 1/E(X).

重要注意：一般情况下E(X²) ≠ [E(X)]²，且E(1/X) ≠ 1/E(X)。

Example: If P(X = 1) = 0.4 and P(X = 2) = 0.6, then E(X²) = 1² × 0.4 + 2² × 0.6 = 2.8, while [E(X)]² = (1 × 0.4 + 2 × 0.6)² = 1.6² = 2.56.

例如：若P(X = 1) = 0.4，P(X = 2) = 0.6，则E(X²) = 1² × 0.4 + 2² × 0.6 = 2.8，而[E(X)]² = (1 × 0.4 + 2 × 0.6)² = 1.6² = 2.56。

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11. Independent Discrete Variables | 独立离散随机变量

If X and Y are independent discrete random variables, then:

若X与Y是独立离散随机变量，则：

• E(X + Y) = E(X) + E(Y) (always true, even without independence)

  E(X + Y) = E(X) + E(Y)（始终成立，即使不独立）

• E(XY) = E(X)E(Y) (requires independence)

  E(XY) = E(X)E(Y)（需要独立性）

• Var(X + Y) = Var(X) + Var(Y) (requires independence)

  Var(X + Y) = Var(X) + Var(Y)（需要独立性）

These rules are essential when combining independent random variables, such as finding the total number of successes in several binomial experiments.

这些规则在合并独立随机变量时非常重要，例如求多个二项试验的总成功次数。

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12. Common Exam Pitfalls and Tips | 常见考试失分点与技巧

Below are typical mistakes and strategies to avoid them.

以下是典型错误及应对策略。

• Misidentifying the distribution: Check whether the problem describes a simple discrete variable, a binomial, or a uniform distribution before applying formulas.

  错误识别分布类型：在套用公式前，先判断问题是简单离散变量、二项分布还是均匀分布。

• Forgetting that probabilities must sum to 1: Use this to find unknown probabilities in a table.

  忘记概率总和为1：利用这一点求表格中的未知概率。

• Using variance formulas incorrectly: Remember Var(X) = E(X²) − (E(X))², not E(X²) − E(X).

  错误使用方差公式：记住Var(X) = E(X²) − (E(X))²，而不是E(X²) − E(X)。

• Calculator errors: When using binomialcdf, check whether the lower bound is included or excluded. For example, P(X < 3) = P(X ≤ 2).

  计算器使用错误：使用binomialcdf时，检查下界是否包含。例如P(X < 3) = P(X ≤ 2)。

• Rounding too early: Keep exact fractions or full decimal precision until the final answer, especially when computing variance.

  过早四舍五入：在最终答案前保留精确分数或完整小数精度，尤其是在计算方差时。

Practicing past paper questions on discrete random variables will build both speed and accuracy.

通过练习历年真题来提升离散随机变量部分的解题速度与准确度。

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