The continuous uniform distribution | 连续均匀分布

📚 The continuous uniform distribution | 连续均匀分布

In A-Level Mathematics, the continuous uniform distribution is one of the most fundamental continuous probability models. It is used when a random variable can take any value in a fixed interval and no sub-interval is more likely than another of the same length. This article covers the key definitions, formulas, and exam techniques required by the Edexcel specification.

在 A-Level 数学中，连续均匀分布是最基本的连续概率模型之一。当一个随机变量可以在一个固定区间内取任意值，并且任何等长子区间都不比其他等长子区间更可能发生时，就可以使用该分布。本文涵盖了 Edexcel 大纲要求的关键定义、公式和考试技巧。

1. Definition and Parameters | 定义与参数

The continuous uniform distribution is defined over a closed interval [a, b], where a is the minimum possible value and b is the maximum possible value. The parameters a and b are real numbers with a < b. We write X ~ U(a, b) to indicate that X follows a uniform distribution on [a, b].

连续均匀分布定义在闭区间 [a, b] 上，其中 a 是最小可能取值，b 是最大可能取值。参数 a 和 b 是实数且 a < b。我们写作 X ~ U(a, b)，表示 X 在 [a, b] 上服从均匀分布。

All values between a and b are equally likely in the sense that any interval of a given length inside [a, b] has the same probability. The total area under the graph must be 1, so the height of the density function is constant.

a 和 b 之间的所有值在以下意义上是等可能的：[a, b] 内任意给定长度的区间具有相同的概率。图形下的总面积必须为 1，因此密度函数的高度是恒定的。

In Edexcel exam questions, the parameters are often given in the context of a real-world scenario, such as a waiting time or a measurement. You should always identify a and b before attempting any calculation.

在 Edexcel 考试题中，参数通常以实际情境给出，例如等待时间或测量值。在进行任何计算之前，你应当先确定 a 和 b。

2. Probability Density Function | 概率密度函数

f(x) = 1 / (b − a) for a ≤ x ≤ b, and f(x) = 0 otherwise.

The probability density function (pdf) of X ~ U(a, b) is constant over [a, b] and zero elsewhere. Because the total area under the curve is 1, the constant height is 1 / (b − a). A wider interval therefore gives a smaller density height.

X ~ U(a, b) 的概率密度函数在 [a, b] 上为常数，在其他地方为零。由于曲线下总面积为 1，常数高度为 1 / (b − a)。因此区间越宽，密度高度越小。

The pdf is never negative, and although f(x) can be greater than 1 when b − a < 1, this is allowed for continuous distributions because f(x) is not a probability at a point.

概率密度函数永远不会为负，并且当 b − a < 1 时，f(x) 可以大于 1，这对于连续分布是允许的，因为 f(x) 不是某一点的概率。

The graph of f(x) is a horizontal line segment from a to b at height 1 / (b − a), and f(x) = 0 everywhere else. This shape is why the distribution is also called the rectangular distribution.

f(x) 的图形是从 a 到 b、高度为 1 / (b − a) 的水平线段，在其他所有位置 f(x) = 0。这种形状也是该分布被称为矩形分布的原因。

3. Cumulative Distribution Function | 累积分布函数

F(x) = 0 for x < a, F(x) = (x − a) / (b − a) for a ≤ x ≤ b, F(x) = 1 for x > b.

The cumulative distribution function F(x) = P(X ≤ x) gives the probability that X is at most x. For the uniform distribution it grows linearly from 0 at x = a to 1 at x = b.

累积分布函数 F(x) = P(X ≤ x) 表示 X 不超过 x 的概率。对于均匀分布，它从 x = a 处的 0 线性增长到 x = b 处的 1。

For a ≤ x ≤ b, F(x) is the area under the density curve from a to x, which is the rectangle width (x − a) multiplied by height 1 / (b − a). Outside [a, b] it is 0 on the left and 1 on the right.

对于 a ≤ x ≤ b，F(x) 是密度曲线从 a 到 x 的面积，即矩形宽度 (x − a) 乘以高度 1 / (b − a)。在 [a, b] 之外，左侧为 0，右侧为 1。

Sketching F(x) can help with probability questions: it is a straight line from (a, 0) to (b, 1), flat at 0 before a and flat at 1 after b. You can read off probabilities directly from the vertical axis.

绘制 F(x) 的草图有助于解决概率问题：它是一条从 (a, 0) 到 (b, 1) 的直线，在 a 之前恒为 0，在 b 之后恒为 1。你可以直接从纵轴上读出概率。

4. Mean and Variance | 期望与方差

E(X) = (a + b) / 2

Var(X) = (b − a)² / 12

The expected value or mean of a continuous uniform random variable is exactly the midpoint of the interval [a, b]. This follows from symmetry: the distribution balances at the centre.

连续均匀随机变量的期望值或均值恰好是区间 [a, b] 的中点。这源于对称性：分布在中心处平衡。

The variance measures the spread. A wider interval gives a larger variance, and the formula Var(X) = (b − a)² / 12 is given in the Edexcel formula booklet. The standard deviation is the square root: √[(b − a)² / 12] = (b − a) / (2√3).

方差衡量离散程度。区间越宽，方差越大，公式 Var(X) = (b − a)² / 12 在 Edexcel 公式表中给出。标准差是其平方根：√[(b − a)² / 12] = (b − a) / (2√3)。

You may be asked to derive E(X) by integration: E(X) = ∫ₐᵇ x / (b − a) dx. Evaluating this gives (b² − a²) / [2(b − a)] = (a + b) / 2, which confirms the midpoint result.

考试中可能会要求通过积分推导 E(X)：E(X) = ∫ₐᵇ x / (b − a) dx。计算后得到 (b² − a²) / [2(b − a)] = (a + b) / 2，这验证了中点结果。

5. Calculating Probabilities | 计算概率

To find P(c ≤ X ≤ d) for [c, d] inside [a, b], use the constant density: P(c ≤ X ≤ d) = (d − c) / (b − a). This is simply the length of the interval divided by the total length.

对于 [a, b] 内的子区间 [c, d]，要求 P(c ≤ X ≤ d)，使用常数密度：P(c ≤ X ≤ d) = (d − c) / (b − a)。这简化为区间的长度除以总长度。

Because X is continuous, P(X = k) = 0 for any single value k. Therefore P(c < X < d) equals P(c ≤ X ≤ d), and endpoints do not matter.

因为 X 是连续的，对任何单个值 k，P(X = k) = 0。因此 P(c < X < d) 等于 P(c ≤ X ≤ d)，端点不影响概率。

For probabilities outside [a, b], P(X < a) = 0 and P(X > b) = 0. If a probability interval overlaps the boundary, only the part inside [a, b] contributes, so you must truncate the interval before applying the formula.

对于 [a, b] 之外的概率，P(X < a) = 0 且 P(X > b) = 0。如果概率区间跨越边界，只有位于 [a, b] 内的部分有贡献，因此在应用公式之前必须将区间截断。

6. Conditional Probability | 条件概率

Uniform distributions often appear in conditional probability questions. For events A and B, P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

均匀分布常出现在条件概率问题中。对于事件 A 和 B，P(A | B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。

If X ~ U(0, 60) and we know X > 20, then the remaining conditional distribution is uniform on (20, 60). The conditional pdf is 1 / 40 on that interval, because the total remaining length is 40.

如果 X ~ U(0, 60) 且已知 X > 20，那么剩余的条件分布在 (20, 60) 上是均匀的。条件概率密度函数在该区间上为 1 / 40，因为剩余总长度为 40。

This ‘memoryless’ rescaling of the uniform distribution is useful: after conditioning on an interval, the variable is still uniform over the new smaller interval. You can then treat the new interval as the whole sample space for further calculations.

均匀分布的这种条件缩放特性很有用：在某个区间上取条件后，变量在新的较小区间上仍然是均匀分布。然后你可以将新区间视为进一步计算的整体样本空间。

7. Percentiles and Median | 百分位数与中位数

The median of X ~ U(a, b) is the midpoint (a + b) / 2. More generally, the p-th percentile is found by solving F(x) = p, giving x = a + p(b − a).

X ~ U(a, b) 的中位数是中点 (a + b) / 2。更一般地，第 p 百分位数通过解 F(x) = p 得到，即 x = a + p(b − a)。

For example, the lower quartile (p = 0.25) is a + 0.25(b − a), and the upper quartile (p = 0.75) is a + 0.75(b − a).

例如，下四分位数（p = 0.25）为 a + 0.25(b − a)，上四分位数（p = 0.75）为 a + 0.75(b − a)。

Since F(x) is linear, percentiles are equally spaced along [a, b]. This linearity makes uniform distribution calculations straightforward and often allows you to use simple ratio methods.

由于 F(x) 是线性的，百分位数沿 [a, b] 等距分布。这种线性使得均匀分布的计算非常直接，并且通常允许你使用简单的比例方法。

8. The Uniform Distribution in Modelling | 均匀分布在建模中的应用

The continuous uniform distribution is used to model situations where there is no tendency for the variable to favour any particular part of an interval. Examples include the arrival time of a bus if it is equally likely at any minute within a 10-minute window, or the position of a random point on a line.

连续均匀分布用于模拟变量在区间内没有任何偏向某个部分的趋势的情况。例子包括：公交车在 10 分钟窗口内任意一分钟到达的概率相等，或直线上随机点的位置。

It is also used in simulations and random number generation. Many programming languages generate uniform random numbers on [0, 1], and these can be transformed to other distributions.

它也用于模拟和随机数生成。许多编程语言生成 [0, 1] 上的均匀随机数，这些随机数可以变换为其他分布。

In exams, a common context is: ‘A train arrives at a station at a random time between 8:00 and 8:20.’ The variable is modelled as U(0, 20) if measured in minutes after 8:00. Always convert units consistently before writing the distribution.

在考试中，常见情境是：“一列火车在 8:00 到 8:20 之间的随机时刻到达车站。”如果以 8:00 后的分钟数计，该变量可建模为 U(0, 20)。在写出分布之前，务必统一单位。

9. Common Mistakes | 常见错误

Do not confuse the continuous uniform distribution with the discrete uniform distribution. In a discrete uniform distribution, the variable takes finitely many equally likely values, while in a continuous one it takes all real values in an interval.

不要将连续均匀分布与离散均匀分布混淆。在离散均匀分布中，变量取有限个等可能的值，而在连续均匀分布中，变量取区间内的所有实数值。

Another common error is forgetting that P(X = k) = 0 for continuous random variables. Adding or removing a single point does not change a probability, so strict and non-strict inequalities give identical results.

另一个常见错误是忘记对于连续随机变量 P

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The continuous uniform distribution | 连续均匀分布