  This work is concerned with approximating multivariate functions in unbounded
domain by using discrete least-squares projection with random points
evaluations. Particular attention are given to functions with random Gaussian
or Gamma parameters. We first demonstrate that the traditional Hermite
(Laguerre) polynomials chaos expansion suffers from the \textit{instability} in
the sense that an \textit{unfeasible} number of points, which is relevant to
the dimension of the approximation space, is needed to guarantee the stability
in the least square framework. We then propose to use the Hermite/Laguerre {\em
functions} (rather than polynomials) as bases in the expansion. The
corresponding design points are obtained by mapping the uniformly distributed
random points in bounded intervals to the unbounded domain, which involved a
mapping parameter $L$. By using the Hermite/Laguerre {\em functions} and a
proper mapping parameter, the stability can be significantly improved even if
the number of design points scales \textit{linearly} (up to a logarithmic
factor) with the dimension of the approximation space. Apart from the
stability, another important issue is the rate of convergence. To speed up the
convergence, an effective scaling factor is introduced, and a principle for
choosing quasi-optimal scaling factor is discussed. Applications to parametric
uncertainty quantification are illustrated by considering a random ODE model
together with an elliptic problem with lognormal random input.
