University Problem Set #12 ECE 313
of Illinois Page 1 of 1 Spring 2002
Assigned: Wednesday, April 10, 2002
Due: Wednesday, April 17, 2002
Reading: Ross, Chapter 6 (except Sections 6.6 and 6.8)
Noncredit Exercises: Ross, Chapter 6: Problems 1, 10-15, 20-23
Problems:
1. Ross, Chapter 6, Problem 8
2. Ross, Chapter 6, Problem 9
3. We return to the “random chord” of Problem 1 of Problem Set #11. Yet another way of
defining a “random chord” is to choose the midpoint of the chord to be anywhere inside the
circle with equal probability. The chord is, of course, perpendicular to the diameter that
passes through the chosen point. Thus, let the random point (X,Y) be uniformly
distributed on the interior of the circle of unit radius centered at the origin (this region is
called the unit disc — nomenclature that might be familiar to DSPists).
(a) Find the probability that the length L of the random chord is greater than the side of the
equilateral triangle inscribed in the circle.
(b) Express L as a function of the random variable (X, Y) and find the probability density
function for L.
(c) Find the average length of the chord, i.e. find E[L].
4. The jointly continuous random variables X and Y have joint pdf given by
fX,Y(u,v) = {2 exp –(u + v), 0 < u < v < ∞,
0, elsewhere.
(a) Sketch the u-v plane and indicate on it the region over which fX,Y(u,v) is nonzero.
(b) Find the marginal pdfs of X and Y.
(c) Are the random variables X and Y independent ?
(d) Find P{Y > 3X}.
(e) For α > 0, find P{X + Y ≤ α}.
(f) Use the result in part (e) to determine the pdf of the random variable Z = X + Y.
5. The jointly continuous random variables X and Y have joint pdf
fX,Y(u,v) =
1/2, 0 ≤ u < 1, 0 ≤ v < 1, and 0 ≤ u + v < 1
3/2, 0 ≤ u < 1, 0 ≤ v < 1, and 1 ≤ u + v < 2
0, otherwise.
Find fX(u), P{X + Y ≤ 3/2} and P{X2 + Y2 ≥ 1}.
6. Let X and Y denote independent N(0, σ2) variables.
(a) What is the joint pdf fX,Y(u,v) of X and Y?
(b) Sketch the u-v plane and indicate on it the region over which you need to integrate the joint
pdf in order to find P{X2 + Y2 > α2}. Then, compute P{X2 + Y2 > α2}. Hint: read the
Solutions to Problems 4(d) and 6(b) of Problem Set #1 and Problem 1(a) of Problem Set #10.
From here onwards, assume σ2 = 1 so that X and Y are independent unit Gaussian RVs.
(c) Express P{|X| > α} in terms of the complementary unit Gaussian CDF function Q(x), and
use this to write P{|X| > α, |Y| > α} in terms of Q(x). (Remember commas mean intersections).
(d) Sketch the u-v plane and show on it the region over which you must integrate the joint pdf
to find P{|X| > α, |Y| > α}. Compare the sketches in parts (b) and (d) to deduce that
P{|X| > α, |Y| > α} ≤ P{X2 + Y2 > 2α2}.
(e) Show that the inequality of part (d) implies that Q(x) ≤ (1/2)•exp(–x2/2) as was proved
earlier in Problem 1(d) of Problem Set #10.