Homework 2 (Due: February 9)

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ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 2
Due: February 9, 2017

Reading
Madhow: Appendix A, especially section A.3, and Section 3.1.
Problems
  1. Madhow: Problem 3.2
  2. Madhow: Problem 3.4
  3. Let X and Y be independent Gaussian random variables with mean m = 0 and variance σ2 = 1.
    1. Sketch a two-dimensional coordinate system with axes X and Y . Indicate the region R1 = {X > α and Y > α} in that coordinate system; assume that α 0.
    2. Show that Pr{X > α,Y > α} = Q2(α). Note that this is the probability that a point (X,Y ) falls in the region R1.
    3. Now, add the region R2 = {X,Y 0 and X2 + Y 2 > 2α2} to your diagram. How does the region R2 compare to R1 from part (a)?
    4. Show that Pr{X,Y 0,X2 +Y 2 > 2α2} = 1
4 exp(-α2). Note that this is the probability that a point (X,Y ) falls in the region R2.
    5. From the above, show that we can conclude the well known bound
              1       α2
Q(α ) ≤ --exp(- --).
        2       2

  4. Let Xt(ω) be a random process defined on Ω = {ω1,4} having probabibility assignments Pr{ωi} = 1
4 for i = 1, 2, 3, 4. The sample functions are
    Xt (ω1) = t       Xt (ω2) = - t
Xt (ω3) = cos2πt  Xt (ω4) = - cos 2πt

    1. Compute the joint probability Pr{X0(ω) = 1,X1(ω) = 1}.
    2. Compute the conditional probability Pr{X1(ω) = 1|X0(ω) = 0}.
    3. Compute the mean and correlation function of Xt(ω).
  5. Prove the following properties of a random process:
    1. RX(t,t) 0
    2. RX(t,u) = RX(u,t) (symmetry)
    3. |RX(t,u)|≤1
2(RX(t,t) + RX(u,u))
    4. |RX(t,u)|2 R X(t,t) RX(u,u)