  The stochastic time-fractional equation $\partial_t \psi
-\Delta\partial_t^{1-\alpha} \psi = f + \dot W$ with space-time white noise
$\dot W$ is discretized in time by a backward-Euler convolution quadrature for
which the sharp-order error estimate \[ {\mathbb
E}\|\psi(\cdot,t_n)-\psi_n\|_{L^2(\mathcal{O})}^2=O(\tau^{1-\alpha d/2}) \] is
established for $\alpha\in(0,2/d)$, where $d$ denotes the spatial dimension,
$\psi_n$ the approximate solution at the $n^{\rm th}$ time step, and
$\mathbb{E}$ the expectation operator. In particular, the result indicates
optimal convergence rates of numerical solutions for both stochastic
subdiffusion and diffusion-wave problems in one spatial dimension. Numerical
examples are presented to illustrate the theoretical analysis.
