s finite, or X = [Xmin, Xmax] no extra conditions; (b) X has an infinite upper or lower bound: (i) X = [Xmin, +00] condition is An < 0, (ii) X = [-00, Xmax condition is An < 0. (c) X is unbounded, or X = [-00, +00]: (i) n is even condition is An < 0, (ii) n is odd condition are An = 0 and An-1 < 0. The above shows that in the situations corresponding to (a), (b), and (c(ii)), it is possible find the set of Lagrange parameters 20, A1,..., An that define the optimal PDF and its maximal entropy. Except for n <2 (as we shall see), this takes an iterative numerical resolution in which the above assumptions are explicitly introduced from the start in order to ensure convergence. The situation (c(ii)) poses a specific problem. Indeed, if we must set An = 0, we are left with n + 1 equations (Eq. (6.29)) to determine the 6.2 Maximum entropy (continuous source) n Lagrange parameters 20, A1, An-1- In the general case, this problem cannot be solved. However, numerical computation makes it possible to find an upper bound to the entropy under said constraints, and to determine an ad-hoc PDF whose entropy approaches it with arbitrary accuracy. While the exact optimal PDF solution does not exist, it can yet be approximated with very high numerical precision. Consider next two simple cases of interest which correspond to gk(x) = xk in the two situations (b(i)) with n = 1 and Xmin = 0, and (c(i)) with n = 2, respectively. In the first case, the PDF and maximum entropy solutions are: p(x)=exp(20-1+2x), (6.30) (6.31) with the convergence condition 21 < 0. The solution defined in Eq. (6.30) corresponds to the continuous-exponential dis- tribution. It is easily established from the constraints in Eq. (6.23) that do = 1 - In(x) and 21 = - 1/(x), which gives p(x) = (1/(x)) xp(-x/(x)) and Hmax = In(e(x)), as the expected result, see Eq. (6.16). In the second case (n = 2), the PDF and maximum entropy solutions are: p(x) = (6.32) Hmax=1-(20+21(x)+22(x2y) (6.33) where A2 < 0. Compared to the previous case, the analytical computation of the 