that they are positive and not both one, and that 1/α + 1/β = 2. To see this symmetry, simply exchange the rôles of i and −i in the Fourier transform. Shannon entropy bound Taking the limit of this last inequality as α, β → 1 yields the less general Shannon entropy inequality, valid for any base of logarithm, as long as we choose an appropriate unit of information, bit, nat, etc. The constant will be different, though, for a different normalization of the Fourier transform, (such as is usually used in physics, with normalizations chosen so that ħ=1 ), i.e., In this case, the dilation of the Fourier transform absolute squared by a factor of 2 simply adds log(2) to its entropy. Entropy versus variance bounds The Gaussian or normal probability distribution plays an important role in the relationship between variance and entropy: it is a problem of the calculus of variations to show that this distribution maximizes entropy for a given variance, and at the same time minimizes the variance for a given entropy. In fact, for any probability density function on the real line, Shannon's entropy inequality specifies: where H is the Shannon entropy and V is the variance, an inequality that is saturated only in the case of a normal distribution. Moreover, the Fourier transform of a Gaussian probability amplitude function is also Gaussian—and the absolute squares of both of these are Gaussian, too. This can then be used to derive the usual Robertson variance uncertainty inequality from the above entropic inequality, enabling the latter to be tighter than the former. That is (for ħ=1), exponentiating the Hirschman inequality and using Shannon's expression above, Hirschman explained that entropy—his version of entropy was the negative of Shannon's—is a "measure of the concentration of [a probability distribution] in a set of small measure." Thus a low or large negative Shannon entropy means that a considerable mass of the probability distribution is confined to a set of small measure. Note that this set of small measure need not be contiguous; a probability distribution can have several concentrations of mass in intervals of small measure, and the entropy may still be low no matter how widely scattered those intervals are. This is not the case with the variance: variance measures the concentration of mass about the mean of the distribution, and a low variance means that a considerable mass of