📚 The Continuous Uniform Distribution | 连续均匀分布
In Edexcel A-Level Mathematics, the continuous uniform distribution is a fundamental probability model where every outcome in an interval [a, b] is equally likely. Understanding its properties, such as the probability density function, cumulative distribution function, mean, variance, and conditional probabilities, is essential for S2 and beyond.
在 Edexcel A-Level 数学中，连续均匀分布是一种基本的概率模型，其中区间 [a, b] 内的每个结果均具有相同可能性。掌握其概率密度函数、累积分布函数、均值、方差和条件概率等性质，对 S2 及后续学习至关重要。
1. Definition and Probability Density Function | 定义与概率密度函数
A continuous random variable X follows a continuous uniform distribution on the interval [a, b], written X ~ U(a, b), if its probability density function (pdf) is constant over that interval and zero elsewhere. The constant height is determined by the requirement that the total area under the graph equals 1.
如果连续随机变量 X 在区间 [a, b] 上的概率密度函数为常数、在区间外为零，则称 X 服从 [a, b] 上的连续均匀分布，记作 X ~ U(a, b)。恒定的密度高度由曲线下总面积为 1 这一要求确定。
f(x) = 1/(b − a) for a ≤ x ≤ b, and f(x) = 0 otherwise.
The pdf is often called a rectangular distribution because its graph is a horizontal line segment. Any subinterval of length Δx inside [a, b] has probability Δx/(b − a).
该概率密度函数常被称为矩形分布，因为其图形是一条水平线段。在 [a, b] 内，任一长度为 Δx 的子区间对应的概率为 Δx/(b − a)。
The following table summarises the key characteristics of the pdf:
下表汇总了概率密度函数的关键特征：
| Feature | Description |
| Domain | a ≤ x ≤ b, where a and b are real numbers with a < b |
| Height of rectangle | 1/(b − a) |
| Area = 1 | (b − a) × [1/(b − a)] = 1 |
| Notation | X ~ U(a, b) or X ~ Unif(a, b) |
2. Cumulative Distribution Function | 累积分布函数
grating the pdf from the lower bound a up to x. For the uniform distribution this integration yields a piecewise linear function.
累积分布函数 F(x) = P(X ≤ x) 是通过从下界 a 到 x 对概率密度函数积分得到的。对于均匀分布，这一积分产生一个分段线性函数。
F(x) = 0 for x < a,
F(x) = (x − a)/(b − a) for a ≤ x ≤ b,
F(x) = 1 for x > b.
Between a and b the cdf increases linearly from 0 to 1. For any x in the interval, F(x) represents the proportion of the total length that lies to the left of x, which is exactly (x − a)/(b − a).
在 a 与 b 之间，累积分布函数从 0 线性增长到 1。对于区间内的任意 x，F(x) 表示位于 x 左侧的长度占总长度的比例，恰好是 (x − a)/(b − a)。
ctly increasing on [a, b], the median and any percentile can be found by solving F(x) = p.
由于 cdf 在 [a, b] 上连续且严格递增，可通过求解 F(x) = p 得到中位数和任意百分位数。
3. Mean and Variance | 均值与方差
The mean (expected value) of X ~ U(a, b) is simply the midpoint of the interval. This follows from symmetry or from integration of x·f(x).
X ~ U(a, b) 的均值（期望值）就是区间的中点。这可由对称性或对 x·f(x) 积分得出。
E(X) = (a + b)/2
The variance measures the spread; for the uniform distribution it depends on the square of the interval width.
方差衡量分散程度；对于均匀分布，它与区间宽度的平方有关。
Var(X) = (b − a)²/12
The standard deviation is the square root of the variance: SD(X) = √[Var(X)] = (b − a)/√12. These formulas are provided in the Edexcel formula booklet, but you should also be able to derive them using integration.
标准差是方差的平方根：SD(X) = (b − a)/√12。Edexcel 公式册中提供这些公式，但你仍应能通过积分推导。
For example, if X ~ U(2, 8), then E(X) = 5 and Var(X) = (8 − 2)²/12 = 36/12 = 3.
例如，若 X ~ U(2, 8)，则 E(X) = 5，Var(X) = (8 − 2)²/12 = 36/12 = 3。
4. Median and Quartiles | 中位数与四分位数
n is symmetric, the median is also the midpoint: Median = (a + b)/2. More generally, the p‑th percentile (0 < p < 1) is the value xₚ such that F(xₚ) = p.
由于均匀分布是对称的，中位数同样是中点：Median = (a + b)/2。更一般地，第 p 百分位数（0 < p < 1）是满足 F(xₚ) = p 的值 xₚ。
xₚ = a + p(b − a)
tile Q₃ (p = 0.75) is a + 0.75(b − a). The interquartile range IQR = Q₃ − Q₁ = 0.5(b − a).
因此，下四分位数 Q₁ (p = 0.25) 为 a + 0.25(b − a)，上四分位数 Q₃ (p = 0.75) 为 a + 0.75(b − a)。四分位距 IQR = Q₃ − Q₁ = 0.5(b − a)。
These percentile formulas are extremely useful because they avoid integration and rely on the proportional nature of the rectangle.
这些百分位数公式十分有用，因为它们避免了积分，直接利用了矩形的比例性质。
5. Probability Calculations | 概率计算
[c, d] (with a ≤ c < d ≤ b) is the length of that subinterval divided by the total length:
对于连续均匀分布，计算概率通常只需简单的比例。X 落在子区间 [c, d]（a ≤ c < d ≤ b）内的概率等于该子区间的长度除以总长度：
P(c ≤ X ≤ d) = (d − c)/(b − a)
For one‑sided probabilities we use the cdf or the complementary property:
对于单侧概率，我们使用累积分布函数或互补性质：
- P(X > x) = 1 − F(x) = (b − x)/(b − a) for a ≤ x ≤ b
- P(X < x) = F(x) = (x − a)/(b − a) for a ≤ x ≤ b
中文：
- P(X > x) = 1 − F(x) = (b − x)/(b − a)，a ≤ x ≤ b
- P(X < x) = F(x) = (x − a)/(b − a)，a ≤ x ≤ b
as appropriate. Remember that for continuous random variables, P(X = specific value) = 0.
如果感兴趣的区间完全位于 [a, b] 之外，概率相应地是 0 或 1。请记住，对于连续随机变量，P(X = 某个特定值) = 0。
6. Conditional Probabilities | 条件概率
b), the remaining probability mass is uniformly spread over (c, b]. Hence the conditional distribution of X given X > c is also uniform: (X | X > c) ~ U(c, b).
均匀分布能产生简洁的条件概率。如果已知 X > c（a ≤ c < b），剩余的概率质量均匀分布在 (c, b] 上。因此，给定 X > c 时 X 的条件分布也是均匀分布：(X | X > c) ~ U(c, b)。
This leads directly to useful formulas:
这直接导出以下实用公式：
P(X > x | X > c) = (b − x)/(b − c) for c ≤ x ≤ b
P(X < x | X > c) = (x − c)/(b − c) for c ≤ x ≤ b
Similarly, for a two‑sided condition:
类似地，对于双侧条件：
P(c < X < d | e < X < f) = (min(d, f) − max(c, e))/(f − e)
is the length of the conditioning interval, provided a ≤ e < f ≤ b and the overlap is non‑empty.
其中分子为两个区间重叠部分的长度，分母为条件区间的长度，前提是 a ≤ e < f ≤ b 且重叠部分非空。
中文补充：这些公式表明，条件概率本质上仍然是长度之比，大大简化了计算。
7. Mode and Shape | 众数与形状
The continuous uniform distribution is a classic example of a distribution that does not have a unique mode. Every value in (a, b) occurs with the same maximum density, so the mode can be described as the whole interval (a, b).
连续均匀分布是没有唯一众数的典型例子。(a, b) 中的每一个值都以相同的最大密度出现，因此众数可视为整个区间 (a, b)。
The shape is perfectly flat – a rectangle of height 1/(b − a) over the base [a, b]. There is no skew; the distribution is symmetric, which is why the mean, median and centre of the interval all coincide.
形状是完全平坦的——在底部 [a, b] 上方形成一个高度为 1/(b − a) 的矩形。没有偏斜；分布是对称的，这就是均值、中位数和区间中心重合的原因。
Because of its simplicity, the uniform distribution often serves as a benchmark for comparing the spread or shape of other distributions.
由于其简单性，均匀分布常作为比较其他分布离散程度或形状的基准。
8. The Standard Uniform Distribution U(0, 1) | 标准均匀分布 U(0, 1)
When a = 0 and b = 1, we obtain the standard continuous uniform distribution, X ~ ). Its pdf is simply f(x) = 1 for 0 ≤ x ≤ 1, and its cdf is F(x) = x on the unit interval.
当 a = 0、b = 1 时，我们得到标准连续均匀分布 X ~ U(0, 1)。其概率密度函数在 0 ≤ x ≤ 1 上就是 f(x) = 1，累积分布函数为 F(x) = x。
E(X) = 0.5, Var(X) = 1/12, and the median = 0.5. Percentiles are equal to p itself: the p‑th percentile is exactly p.
关键性质包括 E(X) = 0.5，Var(X) = 1/12，中位数 = 0.5。百分位数等于 p 本身：第 p 百分位数恰好是 p。
The standard uniform distribution is fundamental in probability theory because it can be transformed to generate random variables from many other distributions.
标准均匀分布在概率论中具有基础地位，因为它可通过变换生成许多其他分布的随机变量。
9. Linear Transformations | 线性变换
If X ~ d d are constants (with c > 0), then Y also follows a continuous uniform distribution. The new bounds are obtained by plugging a and b into the transformation.
若 X ~ U(a, b) 且定义 Y = cX + d，其中 c 和 d 为常数（c > 0），则 Y 同样服从连续均匀分布。新的边界是将 a 和 b 代入变换得到的。
Y ~ U(ca + d, cb + d)
The mean and variance transform in the usual linear way:
均值和方差以通常的线性方式变换：
E(Y) = c·E(X) + d = c·(a + b)/2 + d
Var(Y) = c²·Var(X) = c²·(b − a)²/12
ce/speed. Recognising that the new variable remains uniform saves time and avoids the need for variable transformation using Jacobians.
当题目给出一个均匀随机变量再定义新变量时（例如行程时间 = 距离/速度），这些结果极其有用。认识到新变量仍服从均匀分布可以节省时间，避免使用雅可比行列式进行变量变换。
10. Modelling with the Continuous Uniform Distribution | 连续均匀分布的建模应用
The uniform distribution models any scenario where an outcome is equally likely to occur anywhere within a specified continuous range. Classic examples include:
均匀分布用于模拟在特定连续范围内每个结果发生可能性均等的任何情境。典型例子包括：
- Waiting times: A bus arrives exactly every 15 minutes, and a passenger arrives at the stop at a random instant. The waiting time W ~ U(0, 15).
- Rounding errors: When a number is rounded to the nearest integer, the error E ~ U(−0.5, 0.5).
- Random arrival within a fixed window: A parcel is delivered between 10:00 and 14:00; the arrival time (in minutes after 10:00) can be modelled as U(0, 240).
中文：
- 等待时间：公交车每 15 分钟准时到达一辆，乘客在随机时刻到达车站。等待时间 W ~ U(0, 15)。
- 舍入误差：当一个数四舍五入到最近的整数时，误差 E ~ U(−0.5, 0.5)。
- 固定时间窗内的随机到达：包裹在 10:00 至 14:00 之间送达；到达时间（以 10:00 后分钟计）可建模为 U(0, 240)。
In such contexts, exam questions often ask for the probability that the waiting time exceeds a certain threshold, or for the mean and variance of the quantity of interest. Recognising the uniform structure lets you write down answers using length ratios without lengthy integrals.
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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