Introduction to Information Theory and Data Compression by D.C. Hankerson;Greg A. Harris;Peter D. Johnson Jr.;Peter D.

By D.C. Hankerson;Greg A. Harris;Peter D. Johnson Jr.;Peter D. Johnson;Darrel Hankerson

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Proof: First we show that f is monotone non-increasing on (0, 1]. Suppose that 0 < x < y ≤ 1. Then f (x) = f (y xy ) = f (y) + f (x/y) ≥ f (y), by (a) and (c). Now we use a standard argument using (c) alone to show that f (x r ) = r f (x) for any x ∈ (0, 1] and any positive rational r . First, using (c) repeatedly, or, if you prefer, proceeding by induction on m, it is straightforward to see that for each x ∈ (0, 1] and each positive integer m, f (x m ) = m f (x). Now suppose that x ∈ (0, 1] and that m and n are positive integers.

By the case already done, we have that E(X k ) = p, k = 1, . . , n. Therefore, n n Xk) = E(X) = E( k=1 n E(X k ) = k=1 p = np. 8 Corollary For 0 ≤ p ≤ 1, and any positive integer n , n k k=1 n k p (1 − p)n−k = np. 7, provided there exists, for each p ∈ [0, 1], a Bernoulli trial with probability p of success. You are invited to ponder the existence of such trials (problem 1 at the end of this section). 7 and the existence of such trials for only n + 1 distinct values of p. Therefore, the result follows from the existence of such trials for rational numbers p between 0 and 1.

Again, unless there is some constraint on the number of these events and their probabilities, it does not seem that encoding these as binary words of fixed length tells us anything about units of information, any more than in the case when the events are associated with the same probabilistic experiment. We realize that this discussion is not wholly convincing; perhaps someone will develop a more compelling way of justifying our setup in the future. For now, let us return to E 1 , . . , E m , pairwise mutually exclusive events with m i=1 P(E i ) = 1.

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