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,σ2) 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 σ is very small? What about if σ is very large? Explain intuitively why the results in these extreme cases make sense.

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

The probability that Alice will send 1 and Bob will receive it correctly is P, and the probability that she sends a 0 and Bob receives it correctly is 1 – p. Bob will receive the 1 correctly only if the noise is not below -½, and he will receive the 0 correctly only if the is not above ½.  

Therefore, the probability of receiving both messages correctly is:

P = P(Noise ≤ ½) x (1 – p) + P(Noise ≥ -½) x p

Since the probability of the noise is normally distributed, then:

P(Noise ≤ ½)  = P(Noise ≥ -½)  

Therefore,  

P(Noise ≤ ½)  = P(N/σ ≤ ½σ) = ½σ

(b) What happens to the result from (a) if σ is very small? What about if σ is very large? Explain intuitively why the results in these extreme cases make sense.

Think about this case intuitively, if the noise level is very low and σ is low, then the probability of understanding the message correctly is very high. If σ is low, then ½σ will be high. On the other hand, if the noise level is very high and σ is high, then the probability of understanding the message correctly will be much lower (½σ will be low). Bob will probably be guessing if its a 1 or a 0.

Just imagine when you are trying to talk with someone and there is a lot of noise, you will not be able to understand all the words correctly if the noise level is too high.

-1/2 +2

step-by-step explanation: