Probe measurements in the scrape-off layer of tokamak plasmas show that the ion saturation current signal is dominated by large amplitude bursts. Analysis of data time series from tokamak a configuration variable of unprecedented duration reveals that both burst amplitudes and waiting times are exponentially distributed. The average burst wave form has an exponential shape with a characteristic time that is independent of the burst amplitude. These results support a novel stochastic model, which suggests that the high particle density in the far scrape-off layer is due to the radial motion of uncorrelated blob-like structures. Predictions of the model describe the broad range of universality of plasma fluctuations in the boundary region of tokamak plasmas.