  We study the average case complexity of a linear multivariate problem
$(\lmp)$ defined on functions of $d$ variables. We consider two classes of
information. The first $\lstd$ consists of function values and the second
$\lall$ of all continuous linear functionals. Tractability of $\lmp$ means that
the average case complexity is $O((1/\e)^p)$ with $p$ independent of $d$. We
prove that tractability of an $\lmp$ in $\lstd$ is equivalent to tractability
in $\lall$, although the proof is {\it not} constructive. We provide a simple
condition to check tractability in $\lall$. We also address the optimal design
problem for an $\lmp$ by using a relation to the worst case setting. We find
the order of the average case complexity and optimal sample points for
multivariate function approximation. The theoretical results are illustrated
for the folded Wiener sheet measure.
