📚 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 = xi, Y = yj) 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 = xi, Y = yj 对应的概率 P(X = xi, Y = yj)。
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 pij = P(X = xi, Y = yj). It satisfies two basic conditions:
离散二元分布的联合概率质量函数记为 pij = P(X = xi, Y = yj)。它必须满足两个基本条件：
-
pij ≥ 0 for every pair (xi, yj).
对每一对取值 (xi, yj)，都有 pij ≥ 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 pij = 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 = xi, Y = yj). 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 = xi, Y = yj)，而行合计与列合计则用于求边缘分布。
| X vs Y | Y = 1 | Y = 2 | Y = 3 | P(X = x) |
| X = 1 | 0.1Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信） CommentsMore posts |
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导