A transmitter sends a signal X and a receiver makes the observation Y = X + Z,

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A transmitter sends a signal X and a receiver makes the observation Y = X + Z, whare Z is a receiver noise that is independent of X and E[X] = E[Z] = 0.
Since the average power of the signal is E[X2] and the average power of the noise is E[Z2], a quality measure for the received signal is the signal - to - noise ratio
A transmitter sends a signal X and a receiver makes

How is T related to the correlation coefficient PX,Y?

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