erval [Xmin,Xmax] It is left as an exer- cise for the reader to show that the mean and standard deviation of a continuous, uniform PDF are: (2.15) (2.16) We see from the above results that for a uniform PDF the mean <x> corresponds to the mid-point (Xmax + Xmin)/2 of the interval [Xmin, Xmax]. This was expected, since all events x in this interval are equally probable. However, the deviation o is different from the half width of the distribution, Ax/2, which was not intuitive! This is because (as defined earlier, and to recall) the variance is the mean value of the square of the difference x - <x>, i.e., o2 = ((x <x>)3. Thus, the quantity 20 = Ax/3 does not define a specific interval where the continuous events x are more likely to be observed, although the contrary is true for most other types of nonuniform PDF. Uniform continuous distributions, which take the shape of square or step functions, are not generally found in the physical world. One remarkable example, however, is provided by the so-called Fermi function in semiconductors while under absolute-zero temperature (T = 0 kelvin). Such a function is the probability f(E) of having an electron at a given energy E. The physics shows that, at zero temperature, f(E) = EF for 0 < E EF and f(E) = 0 for E > EF, where EF is referred to as the Fermi energy level. To take a simpler example, assume that one is able to measure some physical parameter within an absolutely defined range. For instance, the distribution of frequencies F from microwave or sunlight radiation, as analyzed through a perfect, square-shaped bandpass filter (suppose that measurements outside that filter range are not observed or irrelevant to the test). If the filter is not too large (or is sufficiently narrow!), and under some circumstances, then one may, indeed, observe that the measurement probability p(F) is uniformly distributed. In any case, this would not mean that the probability is uniform in the absolute definition, but in a local-domain sense, as defined by the observation window of me