University of Illinois Fall 2006
ECE 413: Problem Set 13
Due: Friday May 8 at the beginning of class.
Reading: Ross, Chapter 6 and 7
This Problem Set contains seven problems
1. The random point (X,Y) is uniformly distributed on the shaded region shown in the
left hand figure below.
(a) Find the marginal pdf fX(u) of the random variable X.
(b) Write down the marginal pdf fY(v) of the random variable Yfrom your answer
to part (a).
(c) Find P{X <Y<2X }.
University Problem Set #13 ECE 313
of Illinois Page 1 of 2 Spring 2002
Assigned: Wednesday, April 17, 2002
Due: Wednesday, April 24, 2002
Reading: Ross, Chapter 6 and Chapter 7
Noncredit Exercises: Ross, Chapter 6: Problems 26, 28-30, 41-43, 51, 54;
Theoretical Exercises: 8, 14, 22, 23, 33;
Chapter 7: Problems 1, 16, 26, 29, 34, 36; Theoretical Exercises: 1, 2, 17, 22, 23, 40
Problems:
1. Let (X, Y) have joint pdf fX,Y(u, v) =
C1–u2–v2, u
2
+v2 < 1,
0, elsewhere.
(a) What is the value of C?
(b) Find P{X2+Y2 < 0.25}.
2. The random point (X,Y) is uniformly distributed on the shaded region shown in the left-
hand figure below.
(a) Find the marginal pdf fX(u) of the random variable X.
(b) Write down the marginal pdf fY(v) of the random variable Y from your answer to part (b).
(c) Find P{X < Y < 2X}.
(d) What is fX|Y(u|α), the conditional pdf of X given that Y = α, if α satisfies 0 < α < 1/2?
What is fX|Y(u|α), the conditional pdf of X given that Y = α, if α satisfies 1/2 < α < 1?
Now, apply the theorem of total probability to compute the unconditional pdf of X from
fX|Y(u|α). Do you get the same answer as in part (a)?
u
1
0.5
1
v
0.5
+
-
1v
I
R
R
1
2
3. Two resistors are connected in series to a one-volt voltage source as shown in the right-
hand diagram above. Suppose that the resistance values R1 and R2 (measured in ohms)
are independent random variables, each uniformly distributed on the interval (0, 1). Find
the pdf fI(a) of the current I (measured in amperes) in the circuit.
4. Let (X, Y) have joint pdf fX,Y(u, v) = {2u, 0 < u < 1, 0 < v < 1,
0, elsewhere.
Find the pdf of Z = X2Y.
5. (Unbelievable but true: this problem is easier than it looks…).
(a) If X is N(0,σ2), use the magic formula in Example 7b, Chapter 5.7 of Ross to show that
X2 has gamma pdf with parameter (1/2,1/2σ2).
(b) Now, suppose that X, Y, and Z are independent N(0,σ2) random variables. Then X2,
Y2, and Z2 are independent gamma random variables with parameter (1/2,1/2σ2). Use the
2. Two resistors are connected in series to a one-volt voltage source as shown in the
righthand diagram above. Suppose that the resistance values R1and R2(measured
in ohms) are independent random variables, each uniformly distributed on the interval
(0,1). Find the pdf fI(a) of the current I(measured in amperes) in the circuit.
3. Xand Ydenote independent standard Gaussian random variables.
(a) What is the joint pdf fX,Y(u, v) of Xand Y?
(b) Sketch the u-vplane and indicate on it the region over which you need to integrate
the joint pdf in order to find P{X 2+Y2>2α2}. Compute P{X 2+Y2>2α2}.
(c) Let Z=X2+Y2. What is the pdf of Z?
(d) 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).
(e) On your sketch of part (b), show the region over which you must integrate the
joint pdf to find P{|X | > α, |Y| > α}. Use your sketch to prove the following
result: P{|X | > α, |Y| > α}< P {X 2+Y2>2α2}for α > 0.
(f) Show that inequality of part (e) implies that Q(x)<1
2exp(−x2/2) for x > 0.
