Introduction to Probability, Second Edition, 2nd Edition

7

Joint distributions

When we first introduced random variables and their distributions in Chapter 3, we noted that the individual distributions of two r.v.s do not tell us anything about whether the r.v.s are independent or dependent. For example, two $\text{Bern}\left(1∕2\right)$ r.v.s X and Y could be independent if they indicate Heads on two different coin flips, or dependent if they indicate Heads and Tails, respectively, on the same coin flip. Thus, although the PMF of X is a complete blueprint for X and the PMF of Y is a complete blueprint for Y, these individual PMFs are missing important information about how the two r.v.s are related.

Of course, in real life, we usually care about the relationship between multiple r.v.s in the same experiment. To give ...