Prove $H(X) = H(Y)$

Asked Sep 28 '19 at 21:28

Active Sep 28 '19 at 21:38

Viewed 38 times

If I have $X$ and $Y$ discrete r.v.'s with outcomes $\{x_1,x_2\}$ and $\{y_1,y_2\}$ and entropies $H(X)$ and $H(Y)$. Let $P_{x|y}(x_1|y_1)=P_{x|y}(x_2,y_2)=0$. Prove that $H(X) = H(Y)$ (entropy theory).

If I come up with example numbers and assign that to variables that will prove it. But I need formal way to prove it.

Thanks in advance!

edited Sep 28 '19 at 21:38

Ernie060

6,073

asked Sep 28 '19 at 21:28

Jared

Hint: use the definition of entropy, ant the property for joint and conditional entropy, $H(X,Y)=H(Y|X)+H(X) = H(X|Y)+H(Y)$ – cjferes Sep 28 '19 at 21:35
How to prove it from here? – Jared Sep 28 '19 at 21:37
Use the definition of entropy $H(X)=-\sum_i p_x(x_i)\log\big(p_x(x_i)\big)$ for conditional entropy, i.e. $H(X|Y)=-\sum_i p_{x|y}(x_i|y_i)\log\big(p_{x|y}(x_i|y_i))$ – cjferes Sep 28 '19 at 21:40
then How to conclude this Hx=Hy? – Jared Sep 28 '19 at 21:58

Prove $H(X) = H(Y)$

0 Answers0