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problem_set3

# problem_set3 - EE 511 Problem Set 3 1 The receiver of a...

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EE 511 Problem Set 3 1. The receiver of a communication system receives random variable Y which is defined as Y = X + N in terms of the input random variable X and the channel noise N . X takes on the values - 1 / 4 and 1 / 4 with P [ X = 1 / 4] = 0 . 6. Let f N ( n ) denote the pdf of the channel noise and let X and N be independent. The receiver must decide for each received Y = y whether the transmitted X was - 1 / 4 or 1 / 4. If N is uniform in ( - 1 / 2 , 1 / 2), (a) determine f Y ( y | X = 1 / 4), f Y ( y | X = - 1 / 4) and f Y ( y ) (b) determine the optimal rule such that the probability of correct decision is maximised. 2. If random variables X and Y are related via Y = g ( X ) where g ( . ) is a monotonically increasing function, show that their CDF’s satisfy F Y [ g ( α )] = F X ( α ). 3. Let X be a random variable with pdf f X ( x ). Define Y = g ( X ) where g ( x ) = x | x | ≤ 2 - 2 x < - 2 2 x > 2 (i) Determine f Y ( y ) in terms of f X ( . ), and (ii) Sketch f Y ( y ) when X is a uniform random variable over the interval [-3,3].

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problem_set3 - EE 511 Problem Set 3 1 The receiver of a...

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