In the present research the original Hurst fractal characteristic was applied to the analysis of the human electroencephalogram under different types of cognitive activities. The results obtained in this work showed that by means of the Hurst exponent the EEG-processes, depending on the cortical areas and the functional state, could be reliably discerned. It was found that maximum values of the Hurst exponent were in sites F7 and F8, Broca's area and symmetrical area in the right inferior frontal region. Interhemispherical asymmetry occurred between sites O1 and O2, occipital areas, and that asymmetry correlated positively with general intellectual abilities. The probable mechanisms, lying at the base of the obtained results are discussed in the paper. Mentioned above results showed that application of the original Hurst exponent to the electroencephalogram analysis is an adequate and effective method of analysis of brain activity.