Continuous Random Variables | 连续随机变量

📚 Continuous Random Variables | 连续随机变量

In A-Level Statistics, a random variable is said to be continuous if it can take any value within a given interval. Unlike discrete variables, where probabilities are assigned to individual points, a continuous random variable has a probability density function (PDF) that defines the likelihood of the variable falling within a particular range. The probability of the variable taking an exact single value is always zero. Mastery of continuous distributions is essential for success in the Edexcel examination, as questions routinely require integration to find probabilities, cumulative distribution functions, medians, expectations and variances.

在A-Level统计中，如果一个随机变量可以在给定区间内取任意值，就称其为连续随机变量。与离散变量将概率分配给各个点不同，连续随机变量通过概率密度函数（PDF）来定义变量落在某个范围内的可能性。变量恰好取某个精确值的概率始终为零。掌握连续分布是Edexcel考试成功的关键，因为题目通常要求通过积分求概率、累积分布函数、中位数、期望值和方差。

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1. Continuous vs. Discrete Random Variables | 连续与离散随机变量

A discrete random variable can only take a countable number of distinct values, such as the score on a die or the number of heads in coin tosses. Its probability distribution is given by a probability mass function. In contrast, a continuous random variable can take any value in a continuous interval, for example the height of a student or the time taken to run a race. Probabilities for continuous variables are represented by areas under a curve, not by heights of individual bars.

离散随机变量只能取可数个不同的值，例如骰子的点数或抛硬币的正面向上的次数。它的概率分布由概率质量函数给出。相反，连续随机变量可以在一个连续区间内取任意值，例如学生的身高或跑完比赛所需的时间。连续变量的概率由曲线下的面积表示，而不是由单个柱形的高度来表示。

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2. Probability Density Function (PDF) | 概率密度函数

The probability density function, denoted f(x), describes the relative likelihood of a continuous random variable X taking a value near x. The function itself is not a probability; instead, the area under f(x) over an interval gives the probability. The total area under the curve over the entire range of X must equal 1. Mathematically,

∫_-∞^∞ f(x) dx = 1

概率密度函数记作 f(x)，描述连续随机变量 X 在 x 附近取值的相对可能性。函数本身不是概率；而是由 f(x) 在某个区间上的面积给出概率。X 全部取值范围内曲线下的总面积必须等于 1。数学上表达为：

∫_-∞^∞ f(x) dx = 1

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3. Validating a Probability Density Function | 验证概率密度函数

tire domain equals 1. In many Edexcel questions, you are given a function defined on a finite interval with an unknown constant k. You must set up the integral ∫ f(x) dx = 1 solve for k, and then check that f(x) remains nonnegative on that interval.

要使函数 f(x) 成为一个有效的 PDF，必须满足两个条件。首先，在其定义域内所有 x 满足 f(x) ≥ 0。其次，f(x) 在整个定义域上的积分等于 1。在许多 Edexcel 题目中，你会得到一个在有限区间上定义的函数，且含有未知常数 k。你需要建立积分 ∫ f(x) dx = 1 的方程求解 k，然后检验在该区间上 f(x) 是否保持非负。

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4. Calculating Probabilities Using the PDF | 使用PDF计算概率

Once the PDF is known, the probability that X lies between two values a and b is found by integrating f(x) from a to b:

P(a < X < b) = &#8747;_a^b f(x) dx

< X < b). The integral must be evaluated carefully over the relevant part of the domain.

一旦得到 PDF，X 落在两个值 a 和 b 之间的概率就可以通过对 f(x) 从 a 到 b 积分求得：

P(a < X < b) = &#8747;_a^b f(x) dx

对于连续分布，包含或不包含端点不会改变概率，因为 P(X = a) = 0。因此 P(a ≤ X ≤ b) = P(a < X < b)。必须小心地在相关的定义域部分计算积分。

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5. Cumulative Distribution Function (CDF) and Its Link to PDF | 累积分布函数及其与PDF的关系

The cumulative distribution function F(x) gives the probability that X takes a value less than or equal to x. It is defined as the integral of the PDF up to x:

F(x) = P(X ≤ x) = &#8747;_-∞^x f(t) dt

Conversely, the PDF is the derivative of the CDF wherever F is differentiable: f(x) = F' iven PDF by integrating piecewise, and then to verify consistency by differentiating back.

累积分布函数 F(x) 给出 X 取值小于或等于 x 的概率。它定义为 PDF 积分到 x：

F(x) = P(X ≤ x) = &#8747;_-∞^x f(t) dt

反过来，只要 F 可微，PDF 就是 CDF 的导数：f(x) = F'(x)。Edexcel 题目常要求你通过对给定的 PDF 分段积分求出 CDF，然后通过微分回推验证一致性。

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6. Finding Probabilities from the CDF | 从CDF求概率

(a). To find P(X > a), you can use 1 - F(a), provided a is within the range where F is defined. Always pay attention to piecewise definitions and ensure that you use the correct expression for F(x).

当已知 CDF 时，可直接计算区间上的概率而无需再积分。例如，P(a < X ≤ b) = F(b) - F(a)。要求 P(X > a)，可以使用 1 - F(a)，只要 a 在 F 有定义的范围内。务必注意分段定义，确保使用正确的 F(x) 表达式。

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7. Expectation (Mean) of a Continuous Random Variable | 连续随机变量的期望（均值）

The expected value E(X) represents the long-run average of the random variable. For a continuous variable with PDF f(x), it is given by:

E(X) = &#8747; x f(x) dx

where the integral is taken over the entire domain of X. The mean is a measure of central tendency. In Edexcel problems, you typically integrate a polynomial or simple function, often after having found the constant that makes f a valid PDF.

期望值 E(X) 代表随机变量的长期平均值。对于具有 PDF f(x) 的连续变量，它由下式给出：

E(X) = &#8747; x f(x) dx

其中积分遍及 X 的整个定义域。均值是集中趋势的度量。在 Edexcel 考题中，你通常需要对多项式或简单函数积分，往往是在求出使 f 成为有效 PDF 的常数之后进行。

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

The variance Var(X) measures the spread of the distribution around its mean. It is computed using:

Var(X) = E(X²) - [E(X)]²;

where E(X²) = &# d E(X) and E(X²) first, then apply the formula. If the PDF is piecewise, split the integrals accordingly.

方差 Var(X) 衡量分布在其均值周围的分散程度。通过下式计算：

Var(X) = E(X²) - [E(X)]²;

其中 E(X²) = &#8747; x² f(x) dx。标准差是方差的正平方根， σ = √Var(X)。务必展示清晰的计算过程：先求出 E(X) 和 E(X²)，然后代入公式。如果 PDF 是分段定义的，应相应地将积分拆分。

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9. Expectation of a Function g(X) | 函数的期望 E(g(X))

d to find the expected value of a function of X, such as g(X) = X² + 3X. The rule is:

E[g(X)] = &#8747; g(x) f(x) dx

inear, so E(aX + b) = aE(X) + b, but for non-linear functions you must integrate g(x)f(x).

有时你需要求 X 的函数的期望值，例如 g(X) = X² + 3X。其规则为：

E[g(X)] = &#8747; g(x) f(x) dx

这个通用公式可用于直接计算 E(X²)、E(X³) 或线性变换的期望。注意期望具有线性性质，因此 E(aX + b) = aE(X) + b，但对于非线性函数必须对 g(x)f(x) 积分。

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10. Median and Percentiles | 中位数与百分位数

The median m of a continuous distribution splits the total area under the PDF into two equal halves. It is the value such that

F(m) = 0.5   or   ∫_-∞^m f(x) dx = 0.5

More generally, the 100p-th percentile (where 0 < p < 1) satisfies F(x_p wer quartile (p = 0.25). Solve the equation carefully within the correct piece of the CDF.

连续分布的中位数 m 将 PDF 下的总面积分成相等的两半。它是满足以下条件的值：

F(m) = 0.5   或   ∫_-∞^m f(x) dx = 0.5

更一般地，第 100p 百分位数（0 < p < 1）满足 F(x_p) = p。Edexcel 考题经常要求求中位数、上四分位数（p = 0.75）或下四分位数（p = 0.25）。应在 CDF 的正确分段内仔细求解方程。

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11. Mode of a Continuous Distribution | 连续分布的众数

maximises the PDF f(x). To find it, differentiate f(x) and locate critical points where f' gest f(x) value. Also examine endpoints of the domain if they are included. Unlike median, the mode may not be unique; distributions can be bimodal or multimodal. In exam contexts, always verify that your candidate point is a maximum by using the second derivative or a sign table.

众数是使 PDF f(x) 最大的 x 值。为求众数，对 f(x) 求导并找出 f'(x) = 0 的临界点，然后检查哪个给出最大的 f(x) 值。如果定义域包含端点，还需检查端点。与中位数不同，众数可能不唯一；分布可以是双峰或多峰的。在考试情境下，务必通过二阶导数或符号表验证候选点是最大值。

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12. Worked Example: Uniform Distribution and Exam Advice | 示例：均匀分布与考试建议

erval [a, b], where the PDF is f(x) = 1/(b-a) for a ≤ x ≤ b and zero elsewhere. Then

CDF: F(x) = (x - a)/(b - a),   E(X) = (a + b)/2,   Var(X) = (b - a)²/12

This simple model often appears in exam questions. When solving any continuous random variable problem, sketch the PDF if possible, carefully identify the correct limits of integration, and double-check that CDF values start at 0 and end at 1. For the median, always solve using the CDF if available. Practise with Edexcel past papers to become fluent in integrating piecewise functions and handling algebraic constants within PDFs.

考虑在区间 [a, b] 上的连续均匀分布，其 PDF 为 f(x) = 1/(b-a)，a ≤ x ≤ b，其余地方为零。则

CDF: F(x) = (x - a)/(b - a),   E(X) = (a + b)/2,   Var(X) = (b - a)²/12

这个简单模型常出现在考题中。解任何连续随机变量问题时，如有可能，请画出 PDF 草图，仔细确定正确的积分限，并反复检查 CDF 值是否从 0 开始、到 1 结束。求中位数时，如果可用，务必用 CDF 求解。通过练习 Edexcel 历年真题，熟练掌握分段函数积分以及处理 PDF 中的代数常数。

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