4 A binary erasure channel is slown in the figure below If t

4. A binary erasure channel is slown in the figure below. If the output (erasure symbol) is received, the input has been erased. The probability that \"0\" is erased is o. The probability that \"1\" is erased is E. Assume the input is \"0\" with probability p and \"1\" with probability 1-p. (a) What can you say about the input bit if \"1\" is received? (b) What is the probability that the output is (the erasure symbol)? (c Given that the output is \"r\", what is the probability that the input was \"1\"

Solution

(a) If the output \"1\" is received then it is sure shot possibility that the input was also \"1\"

(b) Probability that Input was \"0\" => p

Probability to obtain \"?\" when input was \"0\" => * p

Probability that Input was \"1\" => ( 1 - p )

Probability to obtain \"?\" when input was \"1\" => * ( 1 - p )

Probability to obtain \"?\" => ( * p ) + ( * ( 1 - p ) )

(c) By Bayes Theorem , Probability to obtain \"?\" when input was \"1\" => ( * ( 1 - p ) ) / ( ( * p ) + ( * ( 1 - p ) ) )