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Question

If X and Y are two discrete random variables and H(V) represents the entropy of the variable V, then consider the following relations:Select the correct relations using the codesgiven below.R1:H(X)H(X|Y)R2:H(X,Y)H(X)+H(Y)R3:H(x)H(Y)=H(X|y)H(Y|X)

A
R1 and R3 only
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B
R1 and R2 only
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C
R1 R2 and R3
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D
R2 and R3 only
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Solution

The correct option is A R1 and R3 only
The mutual information I(X;Y) can be given as,I(X;Y)=H(X)H(X|Y)..(1)I(X;Y)=H(Y)H(Y|X)..(2)I(X;Y)=H(X)+H(Y)H(X,Y)..(3)I(X;Y)0..(4)From equations (1) and (4),H(X)H(X|Y)0H(X)H(X|Y) R1 is correctfrom equations(3) and (4),H(X)+H(Y)H(X,Y)0H(X)+H(Y)H(X,Y) R2 is incorrectFrom equations (1),(2) and (4),H(X)H(X|Y)=H(Y|X)H(X)H(Y)=H(X|Y)H(Y|X)R3 is correctS0,only R1 and R3 are correct.

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