Answer to Question #203893 in Statistics and Probability for moeez

Question

A binary transmission system transmits a signal X ( to send a “000 bit; to send a “100 bit). The received signal is Y = X + N where noise N has a zero-mean Gaussian distribution with variance σ 2 . Assume that “000 bits are three times as likely as “100 bits. (i) Find the conditional pdf of Y given the input value: fY (y|X = +1) and fY (y|X = −1). (ii) The receiver decides a “000 was transmitted if the observed value of y satisfies fY (y|X = −1)P[X = −1] > fY (y|X = +1)P[X = +1] and it decides a “100 was transmitted otherwise. Use the results from part a to show that this decision rule is equivalent to: If y < T decide “000; if y <≥ T decide “100 . (iii) What is the probability that the receiver makes an error given that a +1 was transmitted? a −1 was transmitted? Assume σ 2 = 1/16. (iv) What is the overall probability of error?