Joint Probability Mass Function (PMF) Drill 1

What this lecture covers

This recitation problem practices reading a joint PMF given as a scatter of equally likely points in the plane. It builds on earlier material on joint PMFs, conditioning and expectation, showing how to compute conditional expectations and variances quickly using a "center of mass" shortcut instead of grinding through the full definitions each time.

By the end, you'll be able to find conditional expectation and variance from a small joint PMF, and derive the PMF of a new random variable built from two others, such as their minimum or their product.

Key ideas

• Center of mass shortcut: when outcomes are equally likely, a conditional expectation is just the average (midpoint) of the values that remain possible.
• Conditioning restricts the sample space: fixing x to a specific value keeps only the points on that vertical line, and they stay equally likely relative to each other.
• Variance formula: Var(X|Y=y) = E[X^2|Y=y] - (E[X|Y=y])^2, computed the same way as unconditional variance but within the reduced conditional universe.
• Derived random variables from geometry: comparing points against a line (like y = x) tells you which of two variables is the minimum at each point, letting you read off the PMF of min(X,Y) directly.
• Conditioning on an event: an event like X^2 >= Y again restricts the points to a subset (those below a curve), which are then treated as equally likely among themselves.
• Sanity checks matter: after building a PMF table, its probabilities should sum to 1.

Before you watch

• Know the definitions of joint PMF, conditional PMF, expectation and variance.
• Be comfortable computing expectation as a weighted average over discrete outcomes.
• This drill assumes familiarity with conditioning on random variables from an earlier lecture in the course.

Check your understanding

1. Why does conditioning on X=x preserve the relative probabilities among the remaining points?
2. How would you find the PMF of max(X,Y) using the same geometric approach used for min(X,Y)?
3. Why is the conditional variance of X given Y=0 equal to zero in this problem?
4. What's the difference between computing E[XY] unconditionally versus conditioned on the event A?