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b) Use the above consequence to show that for random variables X and Y taking finite many values, H(X+Y) = H(X)+H(Y) only when X and Y are independent.

b) Use the above consequence to show that for random variables X and Y taking finite many values, H(X+Y) = H(X)+H(Y) only when X and Y are independent.

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Please solve only (b) part

Q*: When do we have equality in Jensen's inequality? a) Assume the following a differentiable function f is a strictly concave up function on an interval a, b. That is, for each xo E [a, b], f(x) > f(x0) + f' (xo)(x – xo)holds for every x E [a, b] and the equality holds only when x = x0. Now, for a finite set I, {p;}ie1, {u;}ieI with 0<pi < 1, Eiei Pi = 1 and u; E a, b]. When do we have Eiej Pif(u;) = Diej f(P;u;)? b) Use the above consequence to show that for random variables X and Y taking finite many values, H(X +Y) = H(X)+H(Y) only when X and Y are independent.
Transcribed Image Text:Q*: When do we have equality in Jensen's inequality? a) Assume the following a differentiable function f is a strictly concave up function on an interval a, b. That is, for each xo E [a, b], f(x) > f(x0) + f' (xo)(x – xo)holds for every x E [a, b] and the equality holds only when x = x0. Now, for a finite set I, {p;}ie1, {u;}ieI with 0<pi < 1, Eiei Pi = 1 and u; E a, b]. When do we have Eiej Pif(u;) = Diej f(P;u;)? b) Use the above consequence to show that for random variables X and Y taking finite many values, H(X +Y) = H(X)+H(Y) only when X and Y are independent.
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