📚 IB Mathematics: Distribution of Functions of Continuous Random Variables | IB数学：连续随机变量函数的分布
In IB Mathematics Analysis & Approaches Higher Level, the topic of continuous random variables often concludes with a fundamental question: given the probability density function (PDF) of a random variable X, what is the PDF of Y = g(X), where g is some function? This is called the distribution of a function of a continuous random variable. It appears in Paper 2 and, especially at HL, in Paper 3. Mastery of this topic requires a solid understanding of cumulative distribution functions (CDFs), differentiation, integration, and careful manipulation of inequalities.
在 IB 数学分析与方法高级课程中，连续型随机变量这一主题通常以这样一个核心问题收尾：已知随机变量 X 的概率密度函数（PDF），如何求 Y = g(X) 的 PDF？这就是连续随机变量函数的分布。它在 Paper 2 中出现，尤其在 HL 的 Paper 3 中更为重要。掌握这一主题需要扎实理解累积分布函数（CDF）、微分、积分，以及谨慎处理不等式。
1. Understanding the Problem | 理解问题
A continuous random variable X is described by its PDF, fX(x), which is non-negative and integrates to 1 over its support. When we apply a transformation g to X, the new variable Y = g(X) is also continuous, provided g is not constant on a set of positive probability. Our goal is to find fY(y), the PDF of Y, using only knowledge of fX(x) and the function g.
连续型随机变量 X 由其概率密度函数 fX(x) 描述，该函数非负，且在其支撑区间上积分为 1。当我们对 X 施加变换 g 后，得到的新变量 Y = g(X) 也是连续的，前提是 g 不在一个正概率集合上为常数。我们的目标是仅利用 fX(x) 和函数 g 的知识，求出 Y 的 PDF，即 fY(y)。
Before beginning any calculation, identify the support of X. For example, if X has support [0, 1], then Y = X² has support [0, 1], while Y = −ln X has support [0, ∞). Getting the support of Y wrong is one of the most common reasons students lose marks in IB examinations.
在开始任何计算之前，先确定 X 的支撑区间。例如，若 X 的支撑为 [0, 1]，则 Y = X² 的支撑为 [0, 1]，而 Y = −ln X 的支撑为 [0, ∞)。把 Y 的支撑写错是学生在 IB 考试中最常见的失分原因之一。
2. The Cumulative Distribution Function (CDF) Method | 累积分布函数法
The CDF method is the most general technique for finding the distribution of Y = g(X). It works even when g is not monotonic, such as for Y = X² or Y = sin X over a symmetric interval. The idea is simple: first find the CDF of Y, denoted FY(y), then differentiate it to obtain the PDF fY(y).
CDF 法是求 Y = g(X) 分布的最通用方法。即使 g 不单调，例如 Y = X² 或在对称区间上的 Y = sin X，它也能适用。其思想很简单：先求 Y 的 CDF，记为 FY(y)，然后对其求导得到 PDF fY(y)。
FY(y) = P(Y ≤ y) = P(g(X) ≤ y) = ∫{x : g(x) ≤ y} fX(x) dx
Once FY(y) is known, the PDF is obtained by differentiation:
fY(y) = d/dy FY(y)
This equation holds at every point where the derivative exists. In many IB questions, the derivative is continuous on the support of Y, so no special care is needed at isolated points.
一旦求得 FY(y)，通过求导即可得到 PDF：该方程在导数存在的每一点成立。在许多 IB 考题中，导数在 Y 的支撑区间上连续，因此在孤立点处无需特殊处理。
3. Step-by-Step Procedure for the CDF Method | CDF 方法的分步步骤
The following procedure can be applied systematically to almost any transformation. It is essential to write down every step clearly in the examination, because method marks are often awarded for the correct setup even if the final derivative contains a small algebraic error.
下面的步骤可以系统地应用于几乎任何变换。在考试中，清晰写出每一步至关重要，因为即使最终导数有小的代数错误，正确的设问过程也常常能获得方法分。
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Write down the definition: FY(y) = P(Y ≤ y) = P(g(X) ≤ y).
写出定义：FY(y) = P(Y ≤ y) = P(g(X) ≤ y)。
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Solve the inequality g(x) ≤ y for x in terms of y. The solution is one or more intervals, depending on the shape of g.
解不等式 g(x) ≤ y，用 y 表示 x。根据 g 的形状，解集是一个或多个区间。
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Express P(g(X) ≤ y) as an integral of fX(x) over the intervals found in Step 2.
将 P(g(X) ≤ y) 表示为 fX(x) 在第 2 步所得区间上的积分。
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Differentiate FY(y) with respect to y to obtain fY(y).
对 FY(y) 关于 y 求导，得到 fY(y)。
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State the support of Y explicitly. The PDF is zero outside this support.
明确写出 Y 的支撑区间；在此区间之外，PDF 为零。
4. The Transformation Formula for Monotonic Functions | 单调函数的变换公式
When g is strictly monotonic, there is a faster direct formula. Suppose g is strictly increasing or strictly decreasing on the support of X, and let h = g⁻¹ be the inverse function. Then the PDF of Y is given by the change-of-variables formula:
当 g 严格单调时，存在更快的直接公式。假设 g 在 X 的支撑区间上严格递增或严格递减，并令 h = g⁻¹ 为其逆函数。则 Y 的 PDF 由变量变换公式给出：
fY(y) = fX( h(y) ) × | h′(y) |
If g is strictly increasing, then h′(y) > 0, so the absolute value is unnecessary. If g is strictly decreasing, then h′(y) < 0, and the absolute value ensures that the PDF remains non-negative. This formula is derived from differentiating the CDF using the inverse function rule and the chain rule.
若 g 严格递增，则 h′(y) > 0，绝对值符号可以省略。若 g 严格递减，则 h′(y) < 0，绝对值可以保证 PDF 保持非负。该公式可由利用反函数法则和链式法则对 CDF 求导得到。
It is essential to remember that this formula only applies when h(y) lies in the support of X. The support of Y is the set of y such that h(y) is in the support of X. If this condition is forgotten, the resulting PDF may be correct algebraically but wrong because it includes values outside the valid range.
必须记住，该公式仅在 h(y) 位于 X 的支撑区间内时才适用。Y 的支撑是使 h(y) 落在 X 支撑内的所有 y 的集合。如果忘记这一条件，得到的 PDF 虽然代数上正确，但因包含了有效范围之外的值而是错误的。
5. Worked Example 1: Linear Transformation | 示例 1：线性变换
Let X be uniformly distributed on (0, 1), so fX(x) = 1 for 0 < x < 1. Define Y = 2X + 3. We want to find the PDF of Y.
设 X 在 (0, 1) 上均匀分布，即当 0 < x < 1 时 fX(x) = 1。定义 Y = 2X + 3。我们要求 Y 的 PDF。
Using the CDF method, for any y:
使用 CDF 法，对任意 y：
FY(y) = P(2X + 3 ≤ y) = P(X ≤ (y − 3)/2)
Because X is uniform on (0, 1), P(X ≤ a) = a for 0 < a < 1. Therefore, for 3 < y < 5:
由于 X 在 (0, 1) 上均匀分布，当 0 < a < 1 时 P(X ≤ a) = a。
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