📚 Solving Problems Involving Random Variables | 求解随机变量相关问题
A random variable is a variable whose value depends on the outcome of a random experiment. In Edexcel A-Level Mathematics, solving problems involving random variables requires you to identify the type of variable, choose the correct distribution, and apply rules for expectation, variance, and probability calculations. This article covers discrete and continuous random variables, linear transformations, special distributions, and common problem-solving strategies.
随机变量是其取值依赖于随机试验结果的变量。在爱德思 A-Level 数学中，求解涉及随机变量的问题需要识别变量类型、选择正确分布，并灵活运用期望、方差和概率计算规则。本文涵盖离散型和连续型随机变量、线性变换、常见分布以及解题策略。
1. Discrete Random Variables and Probability Distributions | 离散随机变量与概率分布
A discrete random variable X takes a finite or countable set of values. Its probability distribution is a list of all possible values x together with their probabilities P(X = x). Two conditions must always hold: each probability must satisfy 0 ≤ P(X = x) ≤ 1, and the sum of all probabilities must equal 1, written as Σ P(X = x) = 1.
离散随机变量 X 取有限个或可数个值。它的概率分布列出了所有可能取值 x 及相应概率 P(X = x)。两个条件必须始终成立：每个概率满足 0 ≤ P(X = x) ≤ 1，且所有概率之和等于 1，写作 Σ P(X = x) = 1。
In many exam questions, one probability is unknown and must be found by using the fact that the total probability is 1. For example, if a discrete random variable X has P(X = 1) = 0.2, P(X = 2) = 0.5 and P(X = 3) = k, then 0.2 + 0.5 + k = 1, so k = 0.3. This type of calculation is usually the first step before finding the mean or variance.
在许多考试题中，某一部分概率未知，需要利用总概率为 1 来求出。例如，若离散随机变量 X 满足 P(X = 1) = 0.2，P(X = 2) = 0.5，P(X = 3) = k，则 0.2 + 0.5 + k = 1，因此 k = 0.3。这类计算通常是求均值或方差前的第一步。
You should also be able to identify when a given table is not a valid probability distribution. If any probability is negative or greater than 1, or if the probabilities do not add to 1, the distribution is invalid. Checking these conditions can help you avoid errors in problem solving.
你还需要能够判断一个给定的表格是否为有效概率分布。如果某个概率为负或大于 1，或者概率之和不等于 1，则该分布无效。检查这些条件有助于避免解题错误。
2. Expected Value E(X) and Variance Var(X) | 期望值 E(X) 与方差 Var(X)
The expected value E(X) is the long-run average value of a random variable. For a discrete random variable, it is calculated by multiplying each value by its probability and summing the results. The formula is centred below.
期望值 E(X) 是随机变量的长期平均值。对于离散随机变量，其计算方法是每个取值乘以对应概率并求和。公式居中如下。
E(X) = Σ x P(X = x)
Variance measures the spread of a random variable around its mean. A very useful formula for variance is Var(X) = E(X²) − [E(X)]². To use it, you first calculate E(X) and E(X²), where E(X²) = Σ x² P(X = x). Remember that variance can never be negative.
方差衡量随机变量围绕均值的离散程度。一个非常有用的方差公式是 Var(X) = E(X²) − [E(X)]²。使用该公式时，先分别计算 E(X) 和 E(X²)，其中 E(X²) = Σ x² P(X = x)。请记住方差永远不会为负。
For example, if X takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3, then E(X) = 1 × 0.2 + 2 × 0.5 + 3 × 0.3 = 2.1. Also E(X²) = 1² × 0.2 + 2² × 0.5 + 3² × 0.3 = 4.9, so Var(X) = 4.9 − 2.1² = 0.49. The standard deviation is the square root of the variance, here √0.49 = 0.7.
例如，若 X 取值 1、2、3，对应概率为 0.2、0.5、0.3，则 E(X) = 1 × 0.2 + 2 × 0.5 + 3 × 0.3 = 2.1。同时 E(X²) = 1² × 0.2 + 2² × 0.5 + 3² × 0.3 = 4.9，因此 Var(X) = 4.9 − 2.1² = 0.49。标准差是方差的平方根，这里为 √0.49 = 0.7。
3. Linear Transformations of Random Variables | 随机变量的线性变换
Problems often ask you to find the mean and variance of a transformed variable such as Y = aX + b, where a and b are constants. The expected value and variance follow simple rules: the mean is transformed in the same way as the variable, while the variance is multiplied by a² and is not affected by adding a constant.
题目经常要求你求变换后变量（如 Y = aX + b，其中 a 和 b 为常数）的均值和方差。期望值和方差遵循简单规则：均值与变量作相同变换，而方差要乘以 a²，且加上常数不会影响方差。
E(aX + b) = aE(X) + b and Var(aX + b) = a² Var(X)
For example, if E(X) = 3 and Var(X) = 4, and Y = 2X − 5, then E(Y) = 2 × 3 − 5 = 1 and Var(Y) = 2² × 4 = 16. The standard deviation of Y is 4, since multiplying X by 2 multiplies the standard deviation by |2|. This is especially useful when converting between units or changing scores in a game.
例如，若 E(X) = 3，Var(X) = 4，且 Y = 2X − 5，则 E(Y) = 2 × 3 − 5 = 1，Var(Y) = 2² × 4 = 16。Y 的标准差为 4，因为 X 乘以 2 会使标准差乘以 |2|。这在单位转换或修改游戏分数时尤其有用。
Be careful with standard deviation: if Y = aX + b, then the standard deviation of Y is |a| times the standard deviation of X. Adding b shifts the whole distribution but does not change its spread.
注意标准差的处理：若 Y = aX + b，则 Y 的标准差是 X 的标准差乘以 |a|。加上 b 只会平移整个分布，不会改变离散程度。
4. The Discrete Uniform Distribution | 离散均匀分布
A discrete uniform distribution occurs when a random variable can take n equally likely values. The simplest case is X taking the values 1, 2, 3, …, n, each with probability 1/n. This distribution appears in problems involving fair dice, random selection from a list, or any situation with equally likely outcomes.
当随机变量可以取 n 个等可能值时，就得到离散均匀分布。最简单的情况是 X 取 1、2、3、…、n，每个值的概率均为 1/n。该分布出现在公平骰子、从列表中随机选取等具有等可能结果的问题中。
For a discrete uniform distribution on 1, 2, …, n, the expected value and variance have standard formulae. These can be quoted in the exam or derived from the general definitions, but using the standard results saves time.
对于取值 1、2、…、n 的离散均匀分布，期望值和方差有标准公式。这些公式可以在考试中直接引用，也可以由一般定义推导，但使用标准结果可以节省时间。
E(X) = (n + 1) / 2 and Var(X) = (n² − 1) / 12
For a fair six-sided die, n = 6, so E(X) = (6 + 1) / 2 = 3.5 and Var(X) = (36 − 1) / 12 = 35/12. In a problem, you might be told that a random variable is uniformly distributed over a set of values and asked to find the probability of a particular outcome or a range of outcomes.
对于一个公平的六面骰子，n = 6，所以 E(X) = (6 + 1) / 2 = 3.5，Var(X) = (36 − 1) / 12 = 35/12。在题目中，你可能会看到随机变量在一组值上均匀分布，并被要求求某个结果或某个范围内结果的概率。
5. The Binomial Distribution | 二项分布
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success p. In Edexcel notation, we write X ~ B(n, p), where n is the number of trials and p is the probability of success.
二项分布用于描述固定次数的独立试验中成功次数的分布，每次试验的成功概率 p 相同。在爱德思考试中，我们写作 X ~ B(n, p)，其中 n 为试验次数，p 为成功概率。
The probability of exactly r successes is given by the binomial formula. You must remember that the trials are independent, there are only two possible outcomes labelled success and failure, n is fixed, and p remains constant for each trial.
恰好有 r 次成功的概率由二项公式给出。你必须记住：各次试验相互独立，每次只有成功和失败两种可能结果，n 固定，并且每次试验的 p 保持不变。
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ
The mean and variance of a binomial random variable can be quoted directly: E(X) = np and Var(X) = np(1 − p). For example, if X ~ B(10, 0.3), then E(X) = 3 and Var(X) = 10 × 0.3 × 0.7 = 2.1. These results are often used to check answers or to set up equations for unknown parameters.
二项随机变量的均值和方差可以直接引用：E(X) = np，Var(X) = np(1 − p)。例如，若 X ~ B(10, 0.3)，则 E(X) = 3，Var(X) = 10 × 0.3 × 0.7 = 2.1。这些结果常用于检验答案或建立未知参数的方程。
6. Using Binomial Cumulative Probabilities | 使用二项累积概率
Many binomial problems ask for probabilities such as P(X ≤ k), P(X < k), P(X ≥ k) or P(a ≤ X ≤ b). These require cumulative probabilities, usually found using a calculator or statistical table. It is important to rewrite each inequality so that it only involves P(X ≤ something).
许多二项分布问题要求求 P(X ≤ k)、P(X < k)、P(X ≥ k) 或 P(a ≤ X ≤ b) 等概率。这些需要累积概率，通常使用计算器或统计表求得。重要的是将每个不等式改写为只包含 P(X ≤ 某个值) 的形式。
For a discrete distribution, P(X < k) is the same as P(X ≤ k − 1), and P(X ≥ k) = 1 − P(X ≤ k − 1). Also P(X > k) = 1 − P(X ≤ k). For an interval, P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1). These adjustments are a very common source of errors, so practise them carefully.
对于离散分布，P(X < k) 等价于 P(X ≤ k − 1)，P(X ≥ k) = 1 − P(X ≤ k − 1)。同样 P(X > k) = 1 − P(X ≤ k)。对于区间，P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1)。这些调整是常见的错误来源，需要仔细练习。
For example, if X ~ B(12, 0.4), then P(X < 5) = P(X ≤ 4), and P(X ≥ 7) = 1 − P(X ≤ 6). If a question says 'more than 3 successes', you need P(X > 3) = 1 − P(X ≤ 3). Always check whether the value itself is included in the inequality.
例如，若 X ~ B(12, 0.4)，则 P(X < 5) = P(X ≤ 4)，P(X ≥ 7) = 1 − P(X ≤ 6)。如果题目说“超过 3 次成功”，你需要求 P(X > 3) = 1 − P(X ≤ 3)。始终要检查该值本身是否包含在不等式中。
7. Continuous Random Variables and Probability Density Functions | 连续随机变量与概率密度函数
A continuous random variable can take any value in an interval. Its distribution is described by a probability density function, or pdf, written f(x). A valid pdf must satisfy two conditions: f(x) ≥ 0 for all x, and the total area under the curve must equal 1.
连续随机变量可以取区间内的任意值。其分布由概率密度函数（pdf）描述，写作 f(x)。一个有效的概率密度函数必须满足两个条件：对所有 x 有 f(x) ≥ 0，且曲线下的总面积等于 1。
∫ f(x) dx = 1 over the whole domain
For a continuous random variable, probabilities are found by integration because probability is represented by area under the density curve. The probability that X lies between a and b is the integral of f(x) from a to b.
对于连续随机变量，概率通过积分求得，因为概率表示为密度曲线下的面积。X 落在 a 与 b 之间的概率就是 f(x) 从 a 到 b 的积分。
P(a < X < b) = ∫ₐᵇ f(x) dx
Unlike discrete variables, a continuous random variable has zero probability of taking any single exact value, so P(X = a) = 0. This means that P(a < X < b) and P(a ≤ X ≤ b) are identical for continuous distributions, and you do not need to worry about adjusting endpoints.
与离散变量不同，连续随机变量取任何一个精确值的概率为零，因此 P(X = a) = 0。这意味着对于连续分布，P(a < X < b) 与 P(a ≤ X ≤ b) 完全相同，你无需担心端点调整问题。
8. Cumulative Distribution Functions | 累积分布函数
The cumulative distribution function, or cdf, gives the probability that a continuous random variable is less than or equal to a given value. It is defined as F(x) = P(X ≤ x) and can be found by integrating the probability density function from the lower end of the domain up to x.
累积分布函数（cdf）给出连续随机变量小于或等于某个给定值的概率。它定义为 F(x) = P(X ≤ x)，可以通过从定义域下限到 x 积分概率密度函数来求得。
F(x) = ∫₋∞ˣ f(t) dt
Once the cdf is known, probabilities over intervals can be found by subtracting values of F. Specifically, P(a ≤ X ≤ b) = F(b) − F(a). You may also be asked to find the probability density function by differentiating the cdf, since f(x) = F'(x).
一旦知道了累积分布函数，区间概率就可以通过 F 的值相减求得。具体地，P(a ≤ X ≤ b) = F(b) − F(a)。你也可能需要通过对 cdf 求导来找出概率密度函数，因为 f(x) = F'(x)。
Another common task is finding the median or quartiles of a continuous distribution. The median m satisfies F(m) = 0.5, the lower quartile Q₁ satisfies F(Q₁) = 0.25, and the upper quartile Q₃ satisfies F(Q₃) = 0.75. You solve these equations by setting the cdf equal to the required probability.
另一个常见任务是求连续分布的中位数或四分位数。中位数 m 满足 F(m) = 0.5，下四分位数 Q₁ 满足 F(Q₁) = 0.25，上四分位数 Q₃ 满足 F(Q₃) = 0.75。你需要令 cdf 等于所需概率来解这些方程。
9. The Normal Distribution | 正态分布
The normal distribution is a continuous distribution that is symmetric about its mean. In Edexcel questions, it is written as X ~ N(μ, σ²), where μ is the mean and σ² is the variance. You must be careful to use the variance, not the standard deviation, in this notation.
正态分布是一种关于均值对称的连续分布。在爱德思考试中，它写作 X ~ N(μ, σ²)，其中 μ 为均值，σ² 为方差。必须注意在该记法中用的是方差，而不是标准差。
To calculate normal probabilities, you convert X to the standard normal variable Z, which has mean 0 and standard deviation 1. The standardised value is found by subtracting the mean and dividing by the standard deviation.
要计算正态分布的概率，需要将 X 转换为标准正态变量 Z，其均值为 0，标准差为 1。标准化的值通过减去均值再除以标准差得到。
Z = (X − μ) / σ where Z ~ N(0, 1)
For example, if X ~ N(50, 4²) and you need P(X < 55), first calculate Z = (55 − 50) / 4 = 1.25. Then use a calculator or table to find Φ(1.25), which gives the probability. Many problems also ask you to work backwards: given a probability, find the corresponding value of X or the unknown mean or standard deviation.
例如，若 X ~ N(50, 4²)，需要求 P(X < 55)，首先计算 Z = (55 − 50) / 4 = 1.25。然后使用计算器或表格求 Φ(1.25)，即得到概率。许多题目还要求反向计算：给定概率，求对应的 X 值或未知的均值与标准差。
10. Solving Word Problems and Exam Strategy | 解决应用题与考试策略
When solving a word problem involving random variables, start by identifying the random variable and its distribution. Ask yourself whether the variable is discrete or continuous, what parameters are given, and what probability or statistic is required. Writing down X ~ B(n, p) or X ~ N(μ, σ²) at the start makes your method clear.
求解涉及随机变量的应用题时，首先要识别随机变量及其分布。问自己变量是离散型还是连续型、给定了哪些参数、需要求什么概率或统计量。一开始写出 X ~ B(n, p) 或 X ~ N(μ, σ²) 能使你的解题思路更清晰。
Next, decide which formula or standard result applies. For discrete variables, you may need to use the probability distribution, binomial formula, or cumulative tables. For continuous variables, you may need to integrate the pdf, use the cdf, or standardise to the normal distribution. Always show the key substitution or equation before using a calculator result.
接下来，确定使用哪个公式或标准结果。对于离散变量，你可能需要使用概率分布、二项公式或累积表。对于连续变量，你可能需要积分 pdf、使用 cdf，或标准化到正态分布。在使用计算器结果之前，始终展示关键的代换或方程。
Finally, check that your answer makes sense in context. Probabilities must lie between 0 and 1, variances must be non-negative, and mean values should be within the possible range of the variable. If a question asks for an interpretation, state clearly what your answer means in terms of the original problem.
最后，检查答案在实际情境中是否合理。概率必须在 0 和 1 之间，方差必须非负，均值应位于变量可能取值范围内。如果题目要求解释，请清楚说明你的答案在原问题中的含义。
Common exam mistakes include confusing variance with standard deviation, forgetting to adjust endpoints for discrete cumulative probabilities, and applying binomial conditions when trials are not independent. Careful reading of the phrase ‘less than’, ‘at least’, ‘more than’, and ‘at most’ is essential, and drawing a simple sketch for normal distribution problems often helps avoid sign errors.
常见考试错误包括混淆方差与标准差、忘记调整离散累积概率的端点，以及在试验不独立时错误套用二项分布。仔细阅读“小于”“至少”“超过”“至多”等措辞至关重要，对于正态分布问题画出简单草图常常有助于避免符号错误。
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