Joint Probability Mass Function (PMF) Drill 2

What this lecture covers

This recitation problem builds further fluency with joint PMFs, following on from an earlier drill. Starting from a joint PMF of X and Y plotted on a grid, with some probability masses left as unknowns, it works through marginalization, sketching a conditional PMF, computing a conditional expectation, and reasoning about independence directly from the structure of the table rather than by formula.

By the end, you'll have a fast, intuitive method for computing conditional PMFs, and a way to test or enforce independence (including conditional independence) by comparing relative frequencies across different conditioning values.

Key ideas

• Marginalization: P(X=x) is found by summing the joint PMF over all values of y, or equivalently by adding the probabilities of the disjoint outcomes where x occurs.
• Fast conditional PMF trick: to condition on X=x, sum the numerators of the joint probabilities in that slice to get a new denominator, then keep the original numerators over that new denominator.
• Conditional expectation via symmetry: for a symmetric conditional PMF, the expectation is just the center of mass, no formula needed.
• Testing independence from a table: if conditioning on different values of x changes the relative frequencies of y, then x tells you something about y, so X and Y cannot be independent, regardless of unknown probability values.
• Zero-probability entries matter: an outcome with probability zero changes which values of y are even possible under a given x, which alone can rule out independence.
• Enforcing conditional independence: given that two variables are independent within a restricted event, the relative frequencies of y across different x values (within that event) must match, which pins down otherwise unknown probabilities.

Before you watch

• Be comfortable with joint PMFs, marginal PMFs and conditional PMFs of discrete random variables.
• Know the definition of independence for random variables from earlier in the course.
• This drill assumes you've seen the "Joint Probability Mass Function (PMF) Drill 1" problem or equivalent material on joint PMFs.

Check your understanding

1. Why does a single zero-probability entry in a joint PMF table make it possible to rule out independence without knowing any other value?
2. Explain the shortcut method for computing a conditional PMF in your own words.
3. In part f, why does the answer to an earlier part (the unconditional joint probability) get reused to compute the conditional probability given event B?
4. How would you check whether two random variables are conditionally independent given some event, versus independent overall?