IB Mathematics: Complete Guide to Bivariate Distributions | IB数学：二元分布考点全解

📚 IB Mathematics: Complete Guide to Bivariate Distributions | IB数学：二元分布考点全解

A bivariate distribution describes the simultaneous behaviour of two random variables, X and Y. In IB Mathematics, you need to read joint probability tables, find marginal and conditional distributions, decide when X and Y are independent, and compute covariance and correlation. This guide explains every key idea with formulas and a full worked example.

二元分布用于描述两个随机变量 X 和 Y 的联合行为。在 IB 数学中，你需要会读联合概率表、求边缘分布与条件分布、判断 X 和 Y 是否独立，并计算协方差与相关系数。本文将系统梳理这些考点，并配完整例题。

1. What Is a Bivariate Distribution? | 什么是二元分布

A univariate distribution gives probabilities for a single random variable. A bivariate distribution gives probabilities for two random variables considered together. For discrete variables, the bivariate distribution is a complete list of the probabilities P(X = x_i, Y = y_j) for every possible pair of values.

一元分布描述单个随机变量的概率；二元分布则把两个随机变量放在一起研究。对于离散型变量，二元分布给出所有可能取值对 X = x_i, Y = y_j 对应的概率 P(X = x_i, Y = y_j)。

For continuous variables, a bivariate distribution is described by a joint density function and double integrals. In IB DP questions, however, the most common form is a discrete joint probability table.

对于连续型变量，二元分布需要用联合密度函数和二重积分来描述；但在 IB 考试中，最常见的题型是离散型联合概率表。

2. Joint Probability Mass Function | 联合概率质量函数

For a discrete bivariate distribution, the joint probability mass function is written as p_ij = P(X = x_i, Y = y_j). It satisfies two basic conditions:

离散二元分布的联合概率质量函数记为 p_ij = P(X = x_i, Y = y_j)。它必须满足两个基本条件：

• p_ij ≥ 0 for every pair (x_i, y_j).

  对每一对取值 (x_i, y_j)，都有 p_ij ≥ 0。

• The sum of all joint probabilities must be 1.

  所有联合概率之和必须等于 1。

∑_i ∑_j p_ij = 1

This condition is exactly the analogue of “total probability equals one” in a one-variable distribution.

这个条件与一元分布中“总概率等于 1”的要求完全对应。

3. Joint Probability Tables | 联合概率表

The quickest way to show a discrete bivariate distribution is a two-way table. Each cell gives P(X = x_i, Y = y_j). The row totals and column totals are useful for marginal distributions.

离散二元分布最常用的是双向表格表示。表格中的每个格子给出 P(X = x_i, Y = y_j)，而行合计与列合计则用于求边缘分布。