Lecture 1.1: Joint PMF of two discrete random variables

Introduction

This tutorial provides a comprehensive overview of the joint probability mass function (PMF) of two discrete random variables, as discussed in Lecture 1.1 from IIT Madras. Understanding joint PMFs is crucial in probability and statistics, especially in fields such as data science, where analyzing relationships between variables is essential.

Step 1: Understanding Random Variables

• Define what a random variable is:
  • A variable whose possible values are numerical outcomes of a random phenomenon.
• Distinguish between discrete and continuous random variables:
  • Discrete random variables take on a finite or countable number of values.
  • Continuous random variables can take on an infinite number of values within a given range.

Step 2: Introduction to Joint PMF

• Explain the concept of joint PMF:
  • The joint PMF of two discrete random variables, X and Y, is a function that gives the probability that each of the variables takes on a specific value.
• Formula representation:
  • The joint PMF is denoted as P(X = x, Y = y), where x and y are specific values of the random variables X and Y.

Step 3: Constructing a Joint PMF Table

• Steps to create a joint PMF table:

  1. Identify the discrete values for each random variable.
  2. Collect data or use theoretical probabilities to fill in the probabilities of each pair (x, y).
  3. Ensure that the sum of all probabilities in the table equals 1.

• Example of a joint PMF table:

  |   Y\X   |  x1  |  x2  |
  |---------|------|------|
  |   y1    |  0.1 |  0.2 |
  |   y2    |  0.3 |  0.4 |
  

Step 4: Marginal PMF Calculation

• Define marginal PMF:

  • The marginal PMF of a random variable is the probability distribution of that variable irrespective of the other variable.

• Steps to calculate the marginal PMF:

  1. For X, sum the joint probabilities across all values of Y.
     • Example: P(X = x1) = P(X = x1, Y = y1) + P(X = x1, Y = y2)
  2. For Y, sum the joint probabilities across all values of X.
     • Example: P(Y = y1) = P(X = x1, Y = y1) + P(X = x2, Y = y1)

Step 5: Properties of Joint PMF

• Discuss key properties:
  • Non-negativity: P(X = x, Y = y) ≥ 0 for all x, y.
  • Normalization: The total probability sums to 1.
  • Independence: If X and Y are independent, P(X = x, Y = y) = P(X = x) × P(Y = y).

Conclusion

In this tutorial, we covered the essentials of joint PMFs including definitions, table construction, and calculating marginal PMFs. Understanding these concepts is critical for analyzing relationships between random variables in data science. As a next step, consider practicing by creating your own joint PMF tables with real or simulated data to reinforce your learning.