According to the statistical theory, upper limit of maxima of random variables is infinity., but we suppose that there must be a certain definite value in the usual circumstances. The author analysed this problem from distinguished characters of probability distribution function, and find the fact that there are two critical values, namely for short-time record of irregular phenomena, about 3% reliability and for long-time record 0.5% respectively. When number of samples is small, probability density curve is likely to parabolla, but number increases it resembles asymptotically to Gauss or Rayleigh distribution. So that if we draw a tangent at the point of inflection on these idealized function, the position of the foot of this tangent may be corresponds to the highest value of maxima of short-time record. Next, the idealized distribution function has two maximum curvature. The one of the position of 'those center of maximum curvature indicates the well-known maximum frequency, so that the other too must have physically important meaning as well, and the author find empirically the fact that this corresponds to the highest value for long-time record. This value is 2.62 times as large as the standard deviation of Gauss distribution, and 2.53 times as large as the mean value of Rayleigh distribution. Existence of maxima is able to conceive on the cumulative density curve of at least thousands or tens of thousands of samples.