  Motivated by the study of simultaneous cores, we give three proofs (in
varying levels of generality) for the expected norm of a weight in a highest
weight representation of a complex simple Lie algebra. First, we argue directly
using the polynomial method and the Weyl character formula. Second, we use the
combinatorics of semistandard tableaux to obtain the result in type A. Third,
and most interestingly, we relate this problem to the "Winnie-the-Pooh problem"
regarding orthogonal decompositions of Lie algebras; although this approach
offers the most explanatory power, it applies only to Cartan types other than A
and C. We conclude with computations of many combinatorial cumulants.
