(A simple problem of digital telecommunication) Alice is trying to transmit to B

Question

(A simple problem of digital telecommunication) 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 N(0, 2). If Bob receives a value greater than 0.5 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 makes sense?

Explanation / Answer

(a) Here the probability that Bob understands Alice correctly.

Bob understands Alice correctly when Alice sends 0 and bob receives zero and alice sends 1 and bob receives one.

so if Alice sends 0, we have to find the probability that bob the signal will be less than 0.5 value.

So Pr(Signal X < 0.5 ) = NORM (X < 0.5 ; 0 ; 2)

Z = 0.5/

so p - value = Pr ( Z < 0.5/) = -1 (0.5/ ) where   is standard normal cumulative distribution function.

(b) Here as will decrease than the probability will increase of answer (a). and if is large than probability will decreasein answer(a). Here intuitively also we can understand that if standar deviation of noise is higher than it will give more inaccurate result and if noise is smalled than it will give less accurate result.

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