Fall 2011 Strategic Practice 8: Section 1 (Covariance and Correlation) - Question 4
Let $X_{1}, ..., X_{k},$ be Multinomial with parameters n and $p_{1}, ..., p_{k},$. Use indicator r.v.s to show that $Cov(X_{i}, X_{j})=-np_{i}p_{j}$ for $i\neq j$.
Solution: Consider the story of the Multinomial, where n objects are being placed into categories 1 through k. Let Ii be the indicator r.v. for object i being in category 1, and let Jj be the indicator r.v. for object j being in category 2. Noting that the ith object is categorized independently of the jth object and using the definition of covariance we get our desired result. See iTunes course for more detail.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."