In this work we consider the problem of
approximating the statistics of a given Quantity of Interest (QoI) that
depends on the solution of a linear elliptic PDE defined over a random
domain parameterized by N random variables. The elliptic problem is
remapped on to a corresponding PDE with a fixed deterministic domain. We
show that t he solution can be analytically extended to a well defined
region in C^{N} with respect to the random variables. A sparse
grid stochastic collocation method is then used to compute the mean and
standard deviation of the QoI. Finally, convergence rates for the mean
and variance of the QoI are derived and compared to those obtained in
numerical experiments.