  We report an exact solution for the noise spectrum of spin-current without
charge-current in a spin field effect transistor (SFET). For the SFET with two
leads, it is found that both auto- and cross-correlation functions are needed
to characterize the noise of spin-current. A shot noise is also generated for
the charge-current even though the net charge-current is identically zero. The
noise spectra of spin-current and charge-current depends on the coupling
strength between the SFET and the leads, and they may behave qualitatively
differently. We also present results of noise spectra in the adiabatic regime.
