Lesson 1

When an experiment gives several numerical characteristics, we should study them altogether. Suppose \(\alpha\) and \(\beta\) are two random variables on the same probability space. Their joint distribution tells us the probabilities of events involving the pair \((\alpha,\beta)\). In the discrete case, it is enough to know the values \[p(a,b)=\Pr[\alpha=a, \beta=b].\] From the joint distribution, we can recover the individual, or marginal, distributions: \[\Pr[\alpha=a]=\sum_{b} p(a,b), \qquad \Pr[\beta=b]=\sum_{a} p(a,b).\] The joint distribution contains more information than the two marginal distributions separately. For example, knowing the distribution of height and the distribution of age is not the same as knowing how height and age are related.