You are trying to construct a channel where comments can be exchanged between pe

Question

You are trying to construct a channel where comments can be exchanged between people. Let 0 be a negative comment and 1 be a positive comment, and Pr(X=0) = 0.8 and P(X=1)=0.2. However, because the channel is noisy, the other end receives an additional noise packet on top of the comment, epsilon, which is an independent Gaussian noise with a variance of 0.39 and an average of 0.

- If the new comment with the noise packet is greater than 0.5 then you know it is positive, and vice versa. What is the probability that you are correct in thinking that you can do this? Answers must be in terms of the cumulative distribution function of the standard Gaussian distribution.

- If the new comment with the noise packet is equal to 0.6, what is the probability that the comment received is positive?

- If the new comment with the noise packet is equal to z, what values of z makes it more likely that you receive a positive comment?

Explanation / Answer

With all three parity bits you can actually correct any one error (out of the 6, 3 message + 3 parity). However, if you only send a4 and a5 then you will be able to correct an error in bit a3 (if a4 and a5s values reflect it), but not any of the others. An error of a1 and an error of a5 are indistinguishable (as with a2 and a4). For error detection, this is still fine, just have the receiver request a resend from the transmitter. But correctable errors reduce retransmissions, speeding up the system.

The probability of an uncorrectable error using all 3 bits is (62)p2(1p)41.5105(62)p2(1p)41.5105

The probability of an undetected error using just 2 bits is (41)p(1p)40.004

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