📚 Modelling with the Continuous Uniform Distribution | 连续均匀分布的建模
The continuous uniform distribution is one of the simplest probability models, yet it plays a vital role in A-Level Statistics when we need to represent a random variable that can take any value within a given interval with equal likelihood. Understanding when and how to model using this distribution is essential for tackling real-world problems and exam questions effectively.
连续均匀分布是最简单的概率模型之一，但在A-Level统计学中，当我们需要表示在给定区间内以等可能性取任意值的随机变量时，它起着至关重要的作用。理解何时以及如何使用此分布进行建模，对于有效解决实际问题和考试题目至关重要。
1. Understanding the Continuous Uniform Distribution | 理解连续均匀分布
The continuous uniform distribution describes a continuous random variable X that can assume any value in the closed interval [a, b], where a and b are constants with a < b. Its probability density function (PDF) is constant across the interval, meaning that every subinterval of equal length has the same probability. We write X ~ U(a, b).
连续均匀分布描述了一个连续随机变量X，它可以取闭区间 [a, b] 中的任何值，其中 a 和 b 是常数且 a < b。其概率密度函数 (PDF) 在整个区间上是恒定的，这意味着每个等长度的子区间具有相同的概率。我们记作 X ~ U(a, b)。
The PDF is given by f(x) = 1/(b - a) for a ≤ x ≤ b, and f(x) = 0 otherwise. The height of the rectangular shape is the reciprocal of the interval width, ensuring the total area under the curve equals 1.
其 PDF 为：当 a ≤ x ≤ b 时，f(x) = 1/(b - a)；否则 f(x) = 0。这个矩形的高度是区间宽度的倒数，以保证曲线下的总面积等于 1。
Because the total area under any valid PDF must be exactly 1, the interval length (b - a) must be finite. An unbounded uniform distribution is not possible. Also, note that the probability of X taking any single exact value is zero, as with all continuous distributions.
由于任何有效PDF下的总面积必须恰好为1，区间长度 (b - a) 必须是有限的。无限区间的均匀分布是不可能的。此外，请注意，与所有连续分布一样，X 取任何单个精确值的概率为零。
2. Cumulative Distribution Function (CDF) | 累积分布函数
ibution builds gradually from 0 to 1 over the interval [a, b]. For values less than a, the probability is 0; for values greater than b, it is 1.
均匀分布的累积分布函数 F(x) = P(X ≤ x) 在区间 [a, b] 上从 0 逐渐增加到 1。对于小于 a 的值，概率为 0；对于大于 b 的值，概率为 1。
Inside the interval, the CDF is linear and can be derived by integrating the constant PDF: F(x) = (x - a) / (b - a) for a ≤ x ≤ b. Thus, the complete definition is: F(x) = 0 for x < a;
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