Modelling with the Continuous Uniform Distribution | 连续均匀分布建模

📚 Modelling with the Continuous Uniform Distribution | 连续均匀分布建模

The continuous uniform distribution is one of the simplest probability models in A-Level Statistics. It describes a continuous random variable for which every interval of the same length within [a, b] has the same probability.

连续均匀分布是 A-Level 统计学中最简单的概率模型之一。它描述一个连续随机变量，在 [a, b] 内任意相同长度的区间都有相同的概率。

1. What Is a Continuous Uniform Distribution? | 什么是连续均匀分布？

A continuous random variable X follows a uniform distribution on the interval [a, b], written X ~ U(a, b), if its probability is spread evenly across the interval. The parameters are the lower bound a and upper bound b, with a < b.

若连续随机变量 X 在区间 [a, b] 上概率均匀分布，记作 X ~ U(a, b)。参数为下界 a 和上界 b，且 a < b。

This means X can take any real value between a and b, but it has no probability outside the interval. The total area under the probability density curve is exactly 1.

这意味着 X 可以取 a 到 b 之间的任意实数值，但在区间之外概率为零。概率密度曲线下的总面积恰好为 1。

2. Probability Density Function | 概率密度函数

The probability density function (PDF) is constant over [a, b] and zero elsewhere.

概率密度函数（PDF）在 [a, b] 内为常数，在区间外为零。

f(x) = 1 / (b − a), a ≤ x ≤ b; f(x) = 0 otherwise

Because the PDF is flat, the graph is a horizontal line between a and b. The height is the reciprocal of the interval width, so a wider interval gives a lower density.

由于 PDF 是平坦的，图像在 a 和 b 之间是一条水平线。高度是区间宽度的倒数，因此区间越宽，密度越低。

3. Cumulative Distribution Function | 累积分布函数

The cumulative distribution function F(x) = P(X ≤ x) increases linearly from 0 to 1 over [a, b].

累积分布函数 F(x)=P(X≤x) 在 [a,b] 内从 0 线性增加到 1。

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

For any value c between a and b, the probability P(X ≤ c) equals the proportion of the interval [a, c] relative to [a, b].

对于 a 与 b 之间的任意 c，概率 P(X≤c) 等于区间 [a,c] 相对于 [a,b] 所占的比例。

A more useful result for modelling is P(c ≤ X ≤ d) = (d − c)/(b − a), provided a ≤ c ≤ d ≤ b.

对于建模更有用的结果是 P(c≤X≤d)=(d−c)/(b−a)，其中 a≤c≤d≤b。

4. Mean, Median and Variance | 均值、中位数与方差

The expected value and median are both the midpoint of the interval, which follows directly from symmetry.

期望值和中位数都是区间中点，这由对称性直接得到。

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

The variance and standard deviation are:

方差和标准差为：

Var(X) = (b − a)² / 12 and σ = √Var(X) = (b − a) / √12

Mean 均值	E(X) = (a + b)/2
Variance 方差	Var(X) = (b − a)² / 12
Standard deviation 标准差	σ = (b − a) / √12

These formulas are given in the Edexcel formula booklet, but you must be able to use them quickly in modelling questions.

这些公式在 Edexcel 公式手册中给出，但你必须在建模题中能够快速使用它们。

5. Core Modelling Assumptions | 核心建模假设

When you model a real situation with a continuous uniform distribution, you assume that the variable is continuous, bounded between a and b, and equally likely in every small subinterval of the same length.

当你用连续均匀分布对实际情况建模时，你假设变量是连续的、在 a 和 b 之间有界，并且在相同长度的每个小区间内等可能。

This assumption of ‘equally likely anywhere in the interval’ is the defining feature of the model. It is reasonable when there is no systematic preference for any particular time or value within the range.

“区间内处处等可能”这一假设是该模型的核心特征。当区间内没有任何特定时间或数值具有系统性偏好时，这个假设是合理的。

For example, if a bus is equally likely to arrive at any time between 10:00 and 10:15, the arrival time can be modelled as U(0, 15) measured in minutes after 10:00.

例如，如果公交车在 10:00 至 10:15 之间任意时刻到达的可能性相同，则到达时间可用 U(0,15) 建模，单位为 10:00 后的分钟数。

6. When to Choose a Uniform Model | 何时选择均匀模型

Uniform models are appropriate in situations such as random number generation, spinner angles, arrival times when people arrive randomly over a fixed period, and measurement rounding errors.

均匀模型适用于以下情况：随机数生成、转盘角度、人们在固定时间段内随机到达的时间，以及测量舍入误差等。

They are usually a first approximation. If data shows values clustering around a central point, a normal model may be better; if events occur over time with no upper bound, an exponential model may be better.

它们通常是一阶近似。如果数据显示数值集中在中心附近，正态模型可能更好；如果事件随时间发生且没有上界，指数模型可能更好。

7. Worked Example: Bus Waiting Time | 例题：公交车等待时间

Example: A bus arrives at a random time between 10:00 and 10:12. Let X be the waiting time in minutes from 10:00 for a passenger who arrives at 10:00. Then X ~ U(0, 12).

例题：公交车在 10:00 至 10:12 之间随机到达。设 X 为从 10:00 起某位乘客的等待时间（分钟），该乘客于 10:00 到达。则 X~U(0,12)。

(a) Find P(X < 3).

（a）求 P(X<3)。

P(X < 3) = 3 / 12 = 1 / 4

(b) Find P(4 < X < 7).

（b）求 P(4<X<7)。

P(4 < X < 7) = (7 − 4) / 12 = 3 / 12 = 1 / 4

（c）求 E(X) 和 Var(X)。

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Modelling with the Continuous Uniform Distribution | 连续均匀分布建模