  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 the 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.
