  We theoretically investigate the fluctuation of a pure spin current induced
by the spin Seebeck effect and spin pumping in a normal metal (NM)/ferromagnet
(FM) bilayer system. Starting with a simple FI--NM interface model with both
spin-conserving and spin-non-conserving processes, we derive general
expressions of the spin current and the spin-current noise at the interface
within second-order perturbation of the FI--NM coupling strength, and estimate
them for an yttrium iron garnet (YIG) --platinum interface. We show that the
spin-current noise can be used to determine the effective spin carried by a
magnon, modified by the spin-non-conserving process at the interface. In
addition, we show that it provides information on the effective spin of a
magnon, heating at the interface under spin pumping, and spin Hall angle of the
NM.
