Consider a discrete memoryless source with alphabet S = {S0, S1,, SK-1} with prob- abilities {Po, P1,..., PK-1}, respectively. The nth extension of this source is another dis- crete memoryless source with source alphabet Sn = {00, σ1,...,σM-1} where M = Κη. Each σ; corresponds to a unique, ordered concatenation of symbols sj₁, Sj2, ..., Sjn from S, j=0, 1, ..., M-1,0j1, j2, Jn K-1. Note and M-1 (a) Show that P(σj) = P(8j₁)P(sj₂) · · · P(Sjn) K-1 K-1 K-1 Σ (63) = ΣΣΣ P(S11) P(812) P (Sin). j=0 j1=0 j2=0 M-1 jn=0 Σ (σ) = 1. j=0 (b) Show that M-1 1 Σ Ρ(σ;) 1082 j=0 Pjk (+) = H(S), k = 1, 2,..., n. (c) Hence, show that M-1 H(S) = P(o) log2 (P(G₁) = nH(S). j=0 (d) Calculate the maximum entropy of an ASCII symbol. (e) Calculate the maximum entropy of an extended ASCII symbol. (f) Do you think the entropy of an English text approaches that of the maximum entropy of an ASCII symbol? Why or why not?