hw8 - EE278 Winter 20102011 Introduction to Statistical...

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EE278 Winter 2010–2011 Introduction to Statistical Signal Processing Problem Set #8 Due: Thursday 10 March 2011 at 5 PM. Last Problem Set 1. Convergence of Random Variables Let X 1 , X 2 , . . . , be iid Exp( λ ) random variables and define four new se- quences for n 1 W n = 1 n n X i = 1 X i V n = min( X 1 , X 2 , . . . , X n ) Y n = n Y i = 1 X i Z n = n × ( W n - 1 ) (a) Does the sequence W n ; n = 1 , 2 , 3 , . . . converge in probability? If so, to what does it converge? (b) Does the sequence V n ; n = 1 , 2 , 3 , . . . converge in probability? If so, to what does it converge? (c) Find the range of λ > 0 such that the sequence Y 1 , Y 2 , Y 3 . . . converges to zero in mean square. (d) Does the sequence Z n ; n = 1 , 2 , 3 , . . . converge in distribution? If so, to what does it converge? You must demonstrate convergence to a particular value, not just state it. 2. Pulse amplitude modulation This problem considers a simple special case of pulse amplitude modulation where a discrete time random process is con- verted into a continuous time random process by multiplying the former by pulses. For simplicity only a binary iid input process is considered. 1
Say that we are given an iid binary random process { X n } with alphabet ± 1, each having probability 1 / 2. We form a continuous time random process { X ( t ) } by assigning X ( t ) = X n ; t [( n - 1) T , nT ) , for a fixed time T . This process can also be described as follows: let p ( t ) be a pulse that is 1 for t [0 , T ) and 0 elsewhere. Define X ( t ) = X k X k p ( t - kT ) . This is an example of pulse amplitude modulation (PAM). Observe that X ( t )