Joint, Marginal, and Conditional Distributions

Learn joint, marginal, and conditional distributions.

26 min read
Advanced

Introduction

Learning Objectives:

  • Work with joint distributions
  • Compute marginals and conditionals
  • Test independence of RVs

Joint PMF

pX,Y(x,y)=P(X=x,Y=y)p_{X,Y}(x,y) = P(X=x, Y=y)

Marginals: pX(x)=sumypX,Y(x,y),quadpY(y)=sumxpX,Y(x,y)p_X(x) = sum_y p_{X,Y}(x,y), quad p_Y(y) = sum_x p_{X,Y}(x,y)

Independence: pX,Y(x,y)=pX(x)cdotpY(y)p_{X,Y}(x,y) = p_X(x) cdot p_Y(y) for all x,yx,y

Key Concepts

This lesson covers fundamental concepts in probability theory.

Key Takeaways

Master these concepts for advanced probability applications.

🎲

Combinatorics and Probability: From Counting to Inference

Master the mathematical foundations of counting, probability, and randomness. This comprehensive course bridges pure combinatorics with computational probability, covering everything from basic counting principles to advanced probabilistic methods, limit theorems, and real-world applications in algorithms, finance, and data science.