MATHEMATICS 2030-Alice is trying to transmit to Bob the answer to a yes-no
Subject: Mathematics / Statistics
Q25, chapter 5, intoduction to probability, Blitzstein. Please solve this question:
Alice is trying to transmit to Bob the answer to a yes-no question, using a noisy channel.
She encodes “yes” as 1 and “no” as 0 , and sends the appropriate value. However, the channel adds noise; specifically, Bob receives what Alice sends plus a N(0, ?2) noise term (the noise is independent of what Alice sends). If Bob receives a value greater than 1/2 he interprets it as “yes”; otherwise, he interprets it as “no”.
(a) Find the probability that Bob understands Alice correctly.
(b) What happens to the result from (a) if ? is very small? What about if ? is very
large? Explain intuitively why the results in these extreme cases make sense.