📚 Joint Probability Density Functions | 联合概率密度函数
In A-Level Further Statistics, joint probability density functions extend the concept of single continuous random variables to two dimensions. This topic is essential for modelling scenarios where two random quantities are observed together, such as height and weight. Understanding joint PDFs lays the foundation for studying covariance, independence, and conditional distributions.
在A-Level进阶统计中，联合概率密度函数将单个连续随机变量的概念推广到二维情形。这一主题对于建模同时观测两个随机量（如身高与体重）的情境至关重要。理解联合概率密度函数为学习协方差、独立性和条件分布奠定基础。
1. Definition of Joint PDF | 联合概率密度函数的定义
A joint probability density function f(x, y) for continuous random variables X and Y is a non-negative function such that the probability of (X, Y) falling in a region A is given by the double integral of f over A. The total volume under the surface must equal 1.
随机变量 X 和 Y 的联合概率密度函数 f(x, y) 是一个非负函数，使得 (X, Y) 落在区域 A 内的概率等于 f 在 A 上的二重积分。曲面下的总体积必须等于 1。
∬ℝ² f(x, y) dx dy = 1
2. Support of a Joint Distribution | 联合分布的支撑集
The support of (X, Y) is the set of all (x, y) for which f(x, y) > 0. It is typically a rectangular region, a triangle, or another bounded domain. Clearly stating the support avoids ambiguity when integrating.
(X, Y) 的支撑集是所有使 f(x, y) > 0 的 (x, y) 的集合。它通常是矩形区域、三角形或其他有界区域。明确写出支撑集可避免积分时的歧义。
3. Marginal Probability Density Functions | 边际概率密度函数
fᵧ(y) = ∫ f(x, y) dx. These functions describe the individual behaviour of each variable.
为获得 X 的边际概率密度函数，需将联合 PDF 对所有可能的 Y 值积分：fₓ(x) = ∫ f(x, y) dy。类似地，fᵧ(y) = ∫ f(x, y) dx。这些函数描述了每个变量各自的分布特性。
4. Checking for Independence | 独立性检验
support: f(x, y) = fₓ(x) · fᵧ(y). If this identity fails at any point, they are dependent.
两个连续随机变量 X 和 Y 相互独立，当且仅当联合 PDF 在整个支撑集上可分解为边际函数的乘积：f(x, y) = fₓ(x) · fᵧ(y)。若该等式在任何一点不成立，则它们不独立。
5. Conditional Probability Density Functions | 条件概率密度函数
The conditional PDF of Y given X = x is defined as f(y| ₓ(x) > 0. This describes the distribution of Y when X is held fixed at a particular value.
给定 X = x 时 Y 的条件概率密度函数定义为 f(y|x) = f(x, y) / fₓ(x)，前提是 fₓ(x) > 0。它描述了当 X 固定在某个特定值时 Y 的分布。
6. Expectation of a Function of Two Variables | 二元函数期望
s obtained by integrating g(x, y) times the joint PDF over the whole support: E[g(X, Y)] = ∬ g(x, y) f(x, y) dx dy. This formula is used for calculating moments, covariance, and more.
对于函数 g(X, Y)，其期望值通过在整个支撑集上对 g(x, y) 与联合 PDF 的乘积进行二重积分获得：E[g(X, Y)] = ∬ g(x, y) f(x, y) dx dy。该公式用于计算矩、协方差等。
7. Covariance and Correlation | 协方差与相关系数
ion coefficient ρ = Cov(X, Y) / (σₓ σᵧ) standardises this measure to lie between −1 and 1.
协方差衡量 X 和 Y 之间的线性关系：Cov(X, Y) = E[(X − μₓ)(Y − μᵧ)] = E[XY] − E[X]E[Y]。相关系数 ρ = Cov(X, Y) / (σₓ σᵧ) 将该度量标准化，使其取值在 −1 到 1 之间。
8. Expectation and Variance of Sums | 求和变量的期望与方差
ar(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y). These linearity properties simplify many proofs.
无论 X 与 Y 是否独立，均有 E[X + Y] = E[X] + E[Y] 以及 Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y)。这些线性性质简化了许多证明。
9. Worked Example: Rectangular Support | 实例：矩形支撑集
uppose f(x, y) = k(2x + y) for 0 < x < 2 and 0 < y < 1, and 0 otherwise. First find k from ∬ f(x, y) dx dy = 1. Compute ∬₀² ∬ x + y) dy dx = k ∫₀² [2xy + y²/2]₀¹ dx = k ∫₀² (2x + ½) dx = k [x² + ½x]₀² = k(4 + 1) = 5k ⇒ k = 1/5.
设 f(x, y) = k(2x + y)，0 < x < 2，0 < y < 1，其余为零。首先由 ∬ f(x, y) dx dy = 1 求 k。计算二重积分得 5k = 1，所以 k = 1/5。
10. Worked Example: Marginal Distributions | 实例：边际分布
Similarly, fᵧ(y) = ∫₀² (1/5)(2x + y) dx = (1/5)[x² + xy]₀² = (1/5)(4 + 2y) for 0 < y < 1.
使用同一个联合 PDF，fₓ(x) = ∫₀¹ (1/5)(2x + y) dy = (4x + 1)/10，0 < x < 2。类似地，fᵧ(y) = (1/5)(4 + 2y)，0 < y < 1。
11. Checking Independence in the Example | 实例中的独立性判断
/5 = (10(2x + y))/50. These are not identically equal (e.g. at x=1, y=0: product gives 5×4/50=0.4, joint gives 20/50=0.4? 5=0.4, product fₓ(1)=0.5, fᵧ(0)=0.2, product=0.1. Not equal). Hence X and Y are dependent.
为判断独立性，考察 f(x, y) 是否等于 fₓ(x) · fᵧ(y)。乘积为 ((4x + 1)/10) × ((4 + 2y)/5) = ((4x + 1)(4 + 2y))/50。而联合 PDF 为 (2x + y)/5 = 10(2x + y)/50。两者不恒等（例如 x=1, y=0 时，乘积为 0.1，联合为 0.4）。因此 X 和 Y 不独立。
12. Summary and Common Pitfalls | 总结与常见错误
Always verify the support before integrating; forgetting the domain is a frequent mistake. Do not assume independence unless the factorisation condition holds. When calculating marginals, ensure the limits of integration correspond to the valid region of the other variable. Practice with triangular and irregular supports to master double integration skills.
积分前务必确认支撑集；忘记定义域是常见错误。除非因子分解条件成立，否则不要假设独立性。计算边际分布时，确保积分限对应于另一个变量的有效区域。通过三角形和不规则支撑集的练习来掌握二重积分技巧。
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