Thermally driven spin-wave spin current in a ferromagnetic material (FM) and the resulting electric signal in a normal metal (NM) probe placed on the FM are theoretically investigated by taking into account the fluctuation-dissipation theorem for thermally fluctuating spin at the interface of an FM-NM junction. We develop a numerical technique for calculating the spin Seebeck signal detected by the NM probe, which converts spin current to charge current by the inverse spin Hall effect. The spin current is induced in the NM probe via an exchange interaction when the FM senses the temperature gradient. Numerical simulation clarifies the role of the sample boundary in the spatial distribution of spin current in both FM and NM.