GATE 2016 EC (EC2) – Question 61
A binary communication system has events $x_0,x_1$ for transmitting zero and one, and $y_0,y_1$ for receiving zero and one. Given $P(x_0)=1/2$, $P(y_0|x_0)=3/4$ and $P(y_0|x_1)=1/2$, the information in bits obtained on learning the received symbol while knowing zero was transmitted is ________.
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Correct answer: 0.80 to 0.82
Explanation
Knowing $x_0$, the conditional probabilities of the received symbols are 3/4 and 1/4. The average information obtained by observing the received symbol is their conditional entropy:
$$H(Y\mid x_0)=-\tfrac34\log_2\tfrac34-\tfrac14\log_2\tfrac14.$$
This evaluates to **0.811278 bits**. The other transmission probability is not needed once the transmitted symbol is specified. The keyed numerical answer represents average information; a particular received outcome would have its own surprisal.