  We consider the estimation of parameter-dependent statistics of functional
outputs of elliptic boundary value problems (BVPs) with parametrized random and
deterministic inputs. For a given value of the deterministic paremeter, a
stochastic Galerkin finite element (SGFE) method can estimate the corresponding
expectation and variance of a linear output at the cost of a single solution of
a large block-structured linear system of equations. We propose a stochastic
Galerkin reduced basis (SGRB) method as a means to lower the computational
burden when statistical outputs are required for a large number of
deterministic parameter queries. Our working assumption is that we have access
to the computational resources necessary to set up such a reduced order model
for a spatial-stochastic weak formulation of the parameter-dependent BVP.
  To derive an SGRB model, we project the spatial-stochastic weak solution of a
parameter-dependent SGFE model onto a POD reduced basis generated from
snapshots of SGFE solutions at representative values of the parameter. We
propose residual-corrected estimates of the parameter-dependent expectation and
variance of linear functional outputs and provide respective computable error
bounds.~We test the SGRB method numerically for a convection-diffusion-reaction
problem, choosing the convective velocity as a deterministic parameter and the
parametrized reactivity field as a random input. Compared to a standard reduced
basis model embedded in a Monte Carlo sampling procedure, the SGRB model
requires a similar number of reduced basis functions to meet a given tolerance
requirement. However, only a single run of the SGRB model suffices to estimate
a statistical output for a new deterministic parameter value, while the
standard reduced basis model must be solved for each Monte Carlo sample.
