Similar to the last example, suppose that X = 0 +W, where O is equally likely...
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Similar to the last example, suppose that X = 0 +W, where O is equally likely to take the values –1 and 1, and where W is standard normal noise, independent of O. We use the estimator 0, with O = 1 if X > 0 and O = -1 otherwise. (This is actually the MAP estimator for this problem.) a) Let us assume that the true value of O is 1. In this case, our estimator makes an error if and only if W has a low (negative) value. The conditional probability of error given the true value of O is 1, that is, P(O +1|0 = 1), is equal to • 4(-1) O (0) O (1) where & is the standard normal CDF. b) For this problem, the overall probability of error is easiest found using the formula O P(Ô + 0) = S P(@ +0|X = x)fx(x) da O P(Ô + 0) =E, P(@ + 0|e = 0) po (0) Similar to the last example, suppose that X = 0 +W, where O is equally likely to take the values –1 and 1, and where W is standard normal noise, independent of O. We use the estimator 0, with O = 1 if X > 0 and O = -1 otherwise. (This is actually the MAP estimator for this problem.) a) Let us assume that the true value of O is 1. In this case, our estimator makes an error if and only if W has a low (negative) value. The conditional probability of error given the true value of O is 1, that is, P(O +1|0 = 1), is equal to • 4(-1) O (0) O (1) where & is the standard normal CDF. b) For this problem, the overall probability of error is easiest found using the formula O P(Ô + 0) = S P(@ +0|X = x)fx(x) da O P(Ô + 0) =E, P(@ + 0|e = 0) po (0)
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Related Book For
Probability and Random Processes With Applications to Signal Processing and Communications
ISBN: 978-0123869814
2nd edition
Authors: Scott Miller, Donald Childers
Posted Date:
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