Sources of tsunamis are non-uniform and commonly uncorrelated and very difficult to predict. The best ideal way to appear their aspects is through heterogeneous or stochastic source models which are more realistic. The effect of random fluctuation of submarine earthquake modeled by vertical time-dependent displacement of a stochastic source model is investigated on the tsunami generation and propagation waves. The noise intensity parameter controls the increase of the stochastic bottom amplitude which results in increasing the oscillations and amplitude in the free surface elevation which provides an additional contribution to tsunami waves. The L<sup>2</sup> norm of the free surface elevation, the displaced water volume and the potential energy are examined. These quantitative information about predicting tsunami risk are useful for risk managers who decide to issue warnings and evacuation orders. The horizontal average velocity flow rates of the tsunami wave are investigated. The average velocity flow rates can provide valuable information about the stochastic bottom topography by the distinctive velocity oscillations. Flow velocity is of importance in risk assessment and hazard mitigation which may provide a clear signal of tsunami flows. Time series of the flow velocities and wave gauges under the effect the water depth of the ocean are investigated.