UNIVERSITY OF SOUTHERN CALIFORNIA, SPRING 2022 1
EE 503: Problem Set #9
Due Thursday Apr. 14, 2022 (9pm)
• Reading: Chapter 5.1-5.8 in Leon-Garcia textbook.
• Submit your homework in D2L by 9pm on the due date.
I. LEON-GARCIA BOOK PROBLEM 5.26ABC + 5.80ABCD (JOINT PDFS OF TWO RVS)
II. INDEPENDENT EXPONENTIAL RANDOM VARIABLES
If X and Y are i.i.d. exponential random variables, find the CDF of Z = XX+Y , and hence its PDF.
III. INDEPENDENT GAUSSIAN RANDOM VARIABLES
Let X and Y be independent identically distributed Gaussian random variables with µ = 0 and σ2 = 4.
(a) Find the joint PDF of X and Y .
(b) Using the joint PDF of part (a), find P [X2 + Y 2 ≤ 1].
(c) If we define Z = X2 + Y 2, find the CDF of Z. What is the name of this distribution?
IV. CONDITIONAL PDF AND ITERATED EXPECTATIONS
The conditional PDF of X given Y is
fX|Y (x|y) = 1√
2π
exp
!
− (x− y)
2
2
"
,
and Y is a uniform random variable in [0, 1].
(a) Find the marginal PDF of X . (You can keep your answer in integral form)
(b) Find E[X] and Var(X).
V. TWO RANDOM VARIABLES AND INDEPENDENCE
The joint CDF of X and Y is
FX,Y =
#
(1− e−αx)(1− e−βy), x ≥ 0, y ≥ 0
0, elsewhere
(a) Find the joint PDF of X and Y .
(b) Find the marginal PDF’s of X and of Y .
(c) Are X and Y independent?
VI. SUM OF RANDOM VARIABLES
X and Y are jointly uniform in the triangular region.
{0 < x < 2} ∩ {0 < y < 1} ∩ {2y < x}.
(a) Find the PDF of Z = X + Y .
(b) Find E[X], E[Y ] and E[XY ].
(c) Find the covariance between X and Y .
UNIVERSITY OF SOUTHERN CALIFORNIA, SPRING 2022 2
VII. CHARACTERISTIC FUNCTION
Let X and Y have the joint PDF
fX,Y (x, y) =
#
ce−(x+y)/2, 0 < x ≤ y <∞
0, otherwise
where c is a constant.
(a) Find the constant c.
(b) Show that the characteristic function of Z = X + Y is
φZ(u) =
1
(1− 2ju)2 , (j =
√−1)
(c) By using the Fourier transform property
jt x(t)←→ d
dω
X(ω)
where X(ω) is the Fourier transform of x(t), find the PDF of Z. Note that
e−αtu(t)←→ 1
α− jω .