By use of Kramer's stationary solution of the Folker-Planck equation for the inverted oscillator, we derive analytic expressions as functions of nuclear temperature and dissipation coefficient for several quantities of interest in the dynamics of nuclear fission. These include the mean time from the saddle point to scission, the mean pre-scission fission-fragment kinetic energy and the contribution to the variance in the fission-fragment kinetic energy resulting from fluctuations in the fission degree of freedom.