We study statistical properties of the random variables Xσ(π),
the number of occurrences of the pattern σ in the permutation π. We
present two contrasting approaches to this problem: traditional probability
theory and the ``less traditional'' computational approach. Through the
perspective of the first one, we prove that for any pair of patterns σ
and τ, the random variables Xσ and Xτ are jointly
asymptotically normal (when the permutation is chosen from Sn). From the
other perspective, we develop algorithms that can show asymptotic normality and
joint asymptotic normality (up to a point) and derive explicit formulas for
quite a few moments and mixed moments empirically, yet rigorously. The
computational approach can also be extended to the case where permutations are
drawn from a set of pattern avoiders to produce many empirical moments and
mixed moments. This data suggests that some random variables are not
asymptotically normal in this setting.Comment: 18 page