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
Madhow: Problem 3.2
Madhow: Problem 3.4
Let X and Y be independent Gaussian random variables with mean
m = 0 and variance σ^{2} = 1.
Sketch a two-dimensional coordinate system with axes X and
Y . Indicate the region R_{1} = {X > α and Y > α} in that
coordinate system; assume that α ≥ 0.
Show that Pr{X > α,Y > α} = Q^{2}(α). Note that this is the
probability that a point (X,Y ) falls in the region R_{1}.
Now, add the region R_{2} = {X,Y ≥ 0 and X^{2} + Y^{2}> 2α^{2}} to
your diagram. How does the region R_{2} compare to R_{1} from
part (a)?
Show that Pr{X,Y ≥ 0,X^{2} +Y^{2}> 2α^{2}} = exp(-α^{2}). Note
that this is the probability that a point (X,Y ) falls in the
region R_{2}.
From the above, show that we can conclude the well known
bound
Let X_{t}(ω) be a random process defined on Ω = {ω_{1},…,ω_{4}} having
probabibility assignments Pr{ω_{i}} = for i = 1, 2, 3, 4. The sample
functions are
Compute the joint probability Pr{X_{0}(ω) = 1,X_{1}(ω) = 1}.
Compute the conditional probability Pr{X_{1}(ω) = 1|X_{0}(ω) =
0}.
Compute the mean and correlation function of X_{t}(ω).
Prove the following properties of a random process: