Joint Distributions: Bivariate Random Variables for Exam P

Joint PMF and PDF

For two discrete random variables X and Y, the joint PMF is p(x, y) = P(X = x, Y = y). For continuous variables, the joint PDF f(x, y) satisfies P((X, Y) in A) = double integral over A of f(x, y) dx dy. In both cases, the function must be non-negative and sum (or integrate) to 1.

On Exam P, a common setup is f(x, y) = c * g(x, y) for (x, y) in some region, and you must find c by integrating. For example, if f(x, y) = c * (x + y) for 0 < x < 1, 0 < y < 1, then integrating gives c * 1 = 1, so c = 1. Actually, the integral of (x + y) over the unit square is 1, so c = 1.

Independence

X and Y are independent if and only if f(x, y) = f_X(x) * f_Y(y) for all x, y. Equivalently, the joint CDF factors: F(x, y) = F_X(x) * F_Y(y). A quick test: if the support region is rectangular and f(x, y) factors into a function of x alone times a function of y alone, then X and Y are independent.

Independence fails when the support region is not rectangular. For instance, if f(x, y) = 2 for 0 < x < y < 1, the triangular support means X and Y cannot be independent, regardless of the form of the density.

Computing Probabilities

To find P(X + Y < 1) for a joint distribution, set up the double integral over the region where x + y < 1. Carefully determine the limits of integration. Drawing the region is essential for avoiding errors. For the density f(x, y) = 2 on 0 < x < y < 1, P(X + Y < 1) = integral from 0 to 1/2 of integral from x to (1-x) of 2 dy dx = 1/2. Practice sketching integration regions, as Exam P heavily rewards this skill.