Hw1_001

# Hw1_001 - ECE 5670 : Digital Communications Spring 2011...

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ECE 5670 : Digital Communications Spring 2011 Homework 1 Due February 3 in class Instructor: Salman Avestimehr Oﬃce 325 Rhodes Hall 1. (a) For a non-negatice integer-valued random variable N , show that E [ N ] = X n =0 Pr( N > n ) (b) For an arbitrary non-negatice random variable X , show that E [ X ] = Z 0 Pr( X a ) da (c) Derive the Markov inequality , which says that for any non-negative random variable X , Pr( X a ) E ( X ) a (d) Derive the Chebyshev inequality , which says that for any random variable X , with ﬁnite mean E [ X ] and ﬁnite variance σ 2 X , Pr( | X - E [ X ] | ≥ b ) σ 2 X b 2 2. Consider the density function f w ( a ) = ± 0 . 5 1 V ≤ | a | ≤ 2 V 0 else. (1) Suppose the energy constraint forces us to transmit voltages in the range ± 2 V . (a) Using the technique from lecture 1, how many distinct voltages can be transmitted so that they are decoded exactly reliably at the receiver? (b) Argue that you can pack more distinct voltages than the previous answer while still

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## This note was uploaded on 10/02/2011 for the course ECE 5670 taught by Professor Scaglione during the Spring '11 term at Cornell.

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Hw1_001 - ECE 5670 : Digital Communications Spring 2011...

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