S/NO answer, which requires a single bit to formulate. In conditions of equiprobability, the uncertainty is evenly distributed between the two events. Such an observation intuitively conveys the sense that the entropy is a maximum in this case. We can immediately verify this by plotting the function H(X) = f(q) = -q logq - (1 - q) log(1-q) defined in Eq. (4.14), from q = 0(x1 impossible) to q = 1(x1 absolutely certain). The plot is shown in Fig. 4.7. As expected, the maximum entropy Hmax = 1 bit is reached for q = 1/2, corresponding to the case of equiprobable events. Formally, the property of entropy maximization can be proved by taking the derivative dH/dq and finding the root, i.e., dH/dq = log[(1 - q)/q] = 0, which yields q = 1/2. We now extend the demonstration to the case of a source with k discrete events, X = {x1, X2: xk} with associated probabilities p(x1), p(x2), p(x). problem of multidimensional optimization, which requires advanced analytical methods (here, namely, the Lagrange multipliers method). The solution is demonstrated in Appendix C. As expected, the entropy is maximum when all the k source events are equiprobable, i.e., p(x1) = p(x2) = = p(x) = 1/k, which yields Hmax = log k. If we assume that k is a power of two, i.e., k = 2M, then Hmax = log 2M = M bits. For instance, a source of 210 = 1024 equiprobable events has an entropy of H = 10 bits. The rest of this chapter, and Appendix C, concerns the issue of PDF optimization for entropy maximization, under parameter constraints. This topic is a bit more advanced than the preceding material. It may be skipped, without compromising the understanding of the following chapters. 4.4 Maximum entropy (discrete source) Probability q Figure 4.7 Plot of the entropy H(X) = f(q) for a source with two complementary events with probabilities q and 1-q (Eq. (4.14)), showing maximum point Hmax = H(0.5) = 1 where events are equiprobable. In Appendix C, I also analyze the problem of entropy maximization through PDF optimization with the introduction of constraint