Alice is trying to transmit to Bob the answer to a yes-no question, using a nois

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, sigma^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". Find the probability that Bob understands Alice correctly. What happens to the result from (a) if sigma is very small? What about if sigma is very large? Explain intuitively why the results in these extreme cases make sense.

Explanation / Answer

So B |(A = 0) N(0, 2 ) and Y |(A = 1) N(1, 2 )

P(Bob interpreted right)

= P(Alice sent 0) * P(Bob Received 0) + P ( Alice sent 1) * P( Bob received 1)

Let the probability that Alice sent 0 be p so P(X=0) = p and P(X=1) = 1-p

P(Bob interpreted right)

= P (B 0.5|A= 0)P(A=0) + P(B > 0.5|B = 1)P(A=1)

Lets calculate P (B 0.5|A = 0); as B |(A = 0) N(0, 2 ) here

So P (B 0.5|A = 0) wil be calculated by Z –value = 0.5/

So P (B 0.5|A = 0) = (0.5/ )

And similarly, P(B > 0.5|B = 1) = (0.5/ )

So P(Bob interpreted right) = p * (0.5/ ) + (1-p) * (0.5/ ) = (0.5/ )

(b) As gets smaller, ( 0.5/ ) gets closer to 1, so the probability that Bob understands Alice correctly gets larger and vice versa.

We can see that if there is vary large variance in noise that means there will be less chance that Bob will hear alice clearly and probbility will go down and viceversa. By intution ,we can say that thing.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.