Further Mathematics: Continuous Random Variables – Core Concepts | 进阶数学：连续随机变量基础概念

📚 Further Mathematics: Continuous Random Variables – Core Concepts | 进阶数学：连续随机变量基础概念

A continuous random variable is one of the central pillars of A-Level Further Statistics. Unlike a discrete random variable which takes isolated values, a continuous random variable can assume any value within an interval, and probabilities are measured over ranges rather than at exact points.

连续随机变量是A-Level进阶统计学的核心支柱之一。与取孤立值的离散随机变量不同，连续随机变量可以在某个区间内取任意值，且概率是在范围上度量，而非在精确点上度量。

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1. From Discrete to Continuous | 从离散到连续

any single value a, because there are infinitely many possible values within any interval.

对于离散随机变量，我们列出每个结果对应的概率，如P(X = 2) = 0.3。然而，对于连续随机变量，P(X = a) = 0，因为任意区间内都存在无穷多个可能取值。

This fundamental difference requires a new tool: the probability density function (PDF), denoted f(x), which describes how probability is distributed across the range of possible values.

这一根本差异要求我们引入新工具：概率密度函数（PDF），记作f(x)，它描述概率在可能取值范围内的分布方式。

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2. The Probability Density Function | 概率密度函数

The probability density function f(x) is a function that describes the relative likelihood of a continuous random variable taking values near a given point. The probability that X lies in an interval [a, b] is found by integrating f(x) from a to b.

概率密度函数f(x)描述连续随机变量在某个点附近取值的相对可能性。X落在区间[a, b]内的概率通过对f(x)从a到b积分求得。

P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx

For Edexcel A-Level Further Mathematics, you must be comfortable interpreting f(x) as a curve, where the area under the curve between two points equals the probability of the variable lying in that interval.

在Edexcel进阶数学考试中，你必须熟练将f(x)理解为一条曲线，曲线上两点之间的面积等于变量落在此区间内的概率。

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3. Validity Conditions of a PDF | 概率密度函数的有效性条件

A function f(x) is a valid probability density function if and only if it satisfies two essential conditions.

函数f(x)是有效的概率密度函数，当且仅当它满足两个基本条件。

• Non-negativity: f(x) ≥ 0 for all x in the sample space.
• Normalisation: The total area under the curve equals 1, i.e., ∫ₐᵇ f(x) dx = 1 over the full range.

• 非负性：对样本空间中的所有x，f(x) ≥ 0。
• 归一化：曲线下总面积为1，即在整个范围内∫ₐᵇ f(x) dx = 1。

These conditions are frequently tested in exam questions asking you to find a constant k in a defined piecewise function.

这些条件在考试中常以求解分段函数中常数k的形式出现。

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4. The Cumulative Distribution Function | 累积分布函数

The cumulative distribution function (CDF), denoted 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 from the lower limit to x.

累积分布函数（CDF），记作F(x)，给出X取值小于或等于x的概率。它定义为PDF从下限到x的积分。

F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt

The CDF is a non-decreasing function that starts at 0 and approaches 1 as x tends to infinity. It connects probabilities directly to cumulative area under the PDF curve.

CDF是非递减函数，从0开始，当x趋向无穷时趋近于1。它将概率直接与PDF曲线下的累积面积联系起来。

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5. Calculating Probabilities from the CDF | 由累积分布函数计算概率

Once the CDF is known, interval probabilities can be computed directly using subtraction.

一旦已知CDF，区间概率可以直接通过减法计算。

P(a < X ≤ b) = F(b) − F(a)

if F(x) = x² for 0 ≤ x ≤ 1, then P(0.2 < X ≤ 0.5) = F(0.5) − F(0.2) = 0.25 − 0.04 = 0.21.

例如，若F(x) = x²（0 ≤ x ≤ 1），则P(0.2 < X ≤ 0.5) = F(0.5) − F(0.2) = 0.25 − 0.04 = 0.21。

ontinuous variables, strict vs non-strict inequalities do not matter: P(X < a) = P(X ≤ a).

记住对于连续变量，严格不等式与非严格不等式没有区别：P(X < a) = P(X ≤ a)。

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6. The Uniform Distribution | 均匀分布

The continuous uniform distribution is the simplest continuous distribution. A random variable X uniformly distributed on [a, b] has constant PDF across its range.

连续均匀分布是最简单的连续分布。在[a, b]上均匀分布的随机变量X在其范围内具有恒定的PDF。

f(x) = 1 ⁄ (b − a), for a ≤ x ≤ b

ppears in Edexcel questions involving random arrival times or random points along a segment.

其CDF是线性的：F(x) = (x − a) / (b − a)，其中a ≤ x ≤ b。该分布常出现在Edexcel考试中涉及随机到达时间或线段上随机点的问题。

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7. Median and Quartiles | 中位数与四分位数

us random variable is the value such that P(X ≤ m) = 0.5. In terms of the CDF, F(m) = 0.5.

连续随机变量的中位数m满足P(X ≤ m) = 0.5。即F(m) = 0.5。

These are special cases of percentiles, where the p-th percentile satisfies F(xₚ) = p/100.

四分位数Q₁和Q₃同理求得：F(Q₁) = 0.25，F(Q₃) = 0.75。它们是百分位数的特殊情况，第p百分位数满足F(xₚ) = p/100。

To find these values in practice, set the CDF equal to the required probability and solve for x. This often involves solving quadratic or cubic equations.

实际求解时，令CDF等于所需概率并解出x。这通常涉及求解二次或三次方程。

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8. Expected Value of a Continuous Random Variable | 连续随机变量的期望值

The expected value, or mean, of a continuous random variable X is the weighted average of all possible values, where the weights are given by the PDF. It is computed via integration.

连续随机变量X的期望值（均值）是所有可能取值的加权平均，权重由PDF给出，通过积分计算。

E(X) = ∫ₐᵇ x f(x) dx

g the intuitive midpoint. For more complex distributions, integration by parts or substitution may be required.

对于[a, b]上的均匀分布，E(X) = (a + b) / 2，符合中点的直观认知。对于更复杂的分布，可能需要分部积分或换元积分。

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

Variance measures the spread of the distribution around the mean. The most efficient formula for continuous variables uses E(X²) and [E(X)]².

方差度量分布围绕均值的离散程度。对连续变量最有效的公式使用E(X²)和[E(X)]²。

Var(X) = E(X²) − [E(X)]²

(x) dx. The standard deviation is simply the square root of the variance: SD(X) = √Var(X).

其中E(X²) = ∫ₐᵇ x² f(x) dx。标准差是方差的平方根：SD(X) = √Var(X)。

Interpreting variance correctly is crucial: a larger variance indicates more spread-out values, while a smaller variance means values cluster closely around the mean.

正确理解方差至关重要：方差越大表示取值越分散，方差越小表示取值紧密集中在均值附近。

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

The mode of a continuous random variable is the value of x at which the PDF f(x) attains its maximum. Unlike the median, which balances probability, the mode locates the most likely region.

连续随机变量的众数是PDF f(x)达到最大值时的x值。与平衡概率的中位数不同，众数定位最可能的区域。

nd derivative is negative to confirm a maximum. If f(x) is defined piecewise, check each interval separately.

求众数时，对f(x)求导并令f′(x) = 0，然后验证二阶导数小于0以确认是极大值。若f(x)为分段函数，需分别检查每个区间。

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11. Transformations of Continuous Random Variables | 连续随机变量的变换

d value of Y can be found without deriving the new PDF, using the law of the unconscious statistician.

当连续随机变量X通过Y = g(X)变换时，无需推导新PDF即可求得Y的期望值，这利用期望的惰性统计学家法则。

E(g(X)) = ∫ₐᵇ g(x) f(x) dx

E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X) are direct consequences of this formula.

这非常有用：E(aX + b) = aE(X) + b以及Var(aX + b) = a²Var(X)都是该公式的直接推论。

In Edexcel exams, you may be asked to find E(X²), E(1/X), or E(eˣ) directly using integration without first finding the distribution of the transformed variable.

在Edexcel考试中，可能要求直接用积分求E(X²)、E(1/X)或E(eˣ)，而不必先求变换后的变量分布。

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12. Exam Strategy and Common Pitfalls | 考试策略与常见陷阱

Success in continuous random variable questions requires both rigorous methodology and awareness of classic mistakes.

要在连续随机变量题目中取得成功，既需要严谨的方法论，也需警惕经典错误。

• y normalisation: check that ∫ f(x) dx = 1 before proceeding with probability calculations.
• Correctly identify the integration limits — many errors arise from integrating outside the defined range of the PDF.
• When finding E(X²), square the variable before integrating; do not integrate first and then square.
• For piecewise functions, ensure each segment of the PDF integrates over its correct domain.
• Sanity-check your final answers: probabilities must lie between 0 and 1, and means should lie within the support of the distribution.

• 始终验证归一化：在进行概率计算前检查∫ f(x) dx = 1。
• 正确确定积分上下限——许多错误源于在PDF定义范围之外积分。
• 求E(X²)时，先平方变量再积分；不要先积分再平方。
• 对于分段函数，确保PDF的每段在其正确定义域内积分。
• 对最终答案做合理性检查：概率必须介于0和1之间，均值应落在分布支撑范围内。

Mastering these core concepts builds a solid foundation for tackling exam questions on continuous probability, from direct PDF calculations to advanced transformation problems.

掌握这些基础概念将为解答连续概率考试题目奠定坚实基础，无论是直接PDF计算还是高级变换问题。

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