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seannasal2804
seannasal2804
06.06.2020 • 
Mathematics

Alice selects one of the numbers +2 and -1 with equal probability and would like to tell you her choice. However, you don't get to see the number itself, but a noisy version that adds to Alice's number a Gaussian random variable with mean 0 and variance o2. Based on this noisy observation, you would like to guess the number Alice originally sent you.(a) Let's say that if your observation is nonpositive, you conclude that the number sent was -1, and if it is positive, that it was +2. Also assume that o = 1. What is the probability of you making an error? (b) Generalizing (a), let's say you decipher Alice's message to be -1 if the observation is below some threshold a (in part (a), we chose a = 0.) What value of a should you use to minimize your probability of error? Assume once again that o2 is unknown.

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