maximum. Using the property P(I** >1)= P(l** 1), we can express this theorem under the alternative form: (8.21) which shows that the probability of the complementary event (l** </+ n) becomes closer to unity as the number of excess bits n increases. Such a condition, however, is not strong enough to guarantee with high probability that 1** (x) < 1(x) in the majority of cases and for any code associated with 1(x). 18 The detailed calculation yields (82) x 2 0.39, (83) X 22 0.48, (84)X 2 0.38, (85) 2 0.47, and (826)x 2 0.33. 19 See T. M. Cover and J.A. Thomas, Elements of Information Theory (New York: John Wiley & Sons, 1991). 8.7 Exercises A second theorem of Shannon codes yields an interesting and quite unexpected prop- erty. It applies to probability distributions for which 1** = l*, meaning that - -log p(x) is an integer for all X. As we have seen earlier, such a distribution is said to be dyadic. This theorem states that for a dyadic source distribution (see proof in previous note): 20 (8.22) where (to recall) l*(x) is the individual codeword length given by Shannon coding and 1(x) is that given by any other code. It is, therefore, always more likely for dyadic sources that the Shannon code yields strictly shorter individual codeword lengths than the opposite. This is indeed a stronger statement than in the previous case, concerning nondyadic sources. It should be emphasized that these two theorems of Shannon-coding optimality concern individual, and not mean codeword lengths. As we shall see in the next chapter, there exist codes for which the mean codeword length is actually shorter than that given by the Shannon code. The different properties that I have established in this section provide a first and vivid illustration of the predictive power of information theory. It allows one to know beforehand the expected minimal code sizes, regardless of the source type and the coding technique (within the essential constraint that the code be a prefix one). This knowledge makes it possible to estimate the efficiency