Course Content
I. Course Information A. Syllabus
II. Problem Sets A. PS 1
B. PS 2
C. PS 3
D. PS 4
E. PS 5
F. PS 6
G. Exercise Problems
H. PS 7
III. Solutions A. Sol 1
B. Sol 2
C. Sol 3
D. hw3.m
E. Sol 4
F. Midterm Solutions
G. Sol 5
H. Sol 6
I. Solutions
ECE 804, Random Signal Analysis
OSU, Autumn 2009
Nov. 23, 2009
Solutions - Problem Set 7
Problem 1
(a)
1
1=
y
fX,Y (x, y) dxdy =
0
y
k
dxdy
y
1
=k
2 dy = 2k.
0
Thus, k = 1 .
2
(b) For 1
x
0,
1
fX (x) =
x
and for 0 < x
1,
1
1
dy = ln(x)
2y
2
1
fX (x) =
x
1
ECE 804, Random Signal Analysis
OSU, Autumn 2009
Nov. 16, 2009
Solutions - Problem Set 6
Problem 1
E (X z)2 = E X 2 2zE [X] + z 2 ,
which is a dierentiable function of z for all z IR. To nd the value, z , of z, which
minimizes the above expression, we die
Solutions - Problem Set 3
Problem 1
(a)
1=
1
fC (c) dc =
kc dc = k
0
Thus, fC (c) =
1
c2
2
=
0
k
2
=
2c, 0 c 1
0, otherwise
(b)
1
P (H) =
k=2
P (H|c) fC (c) dc =
c 2c dc = k
0
2c3
3
1
=
0
2
3
(c)
P (E|C) =
n k
c (1 c)nk
k
1
n k
c (1 c)nk 2c dc
k
0
n (k +
ECE 804, Random Signal Analysis
OSU, Autumn 2009
Oct. 21, 2009
Oct. 28, 2009
Solutions - Problem Set 4
Problem 1
(a) The function Y = f (X) is given by
ag, X a
Y = gX, a X a
+ag, X a
Thus, Y is never less than ga or greater than +ga. Also, for ag
FY (y) =
EE 805, Random Processes and Linear Systems
OSU, Winter 2010
Solutions - PS 1
Problem 1
1
Jan. 18, 2010
Problem 2
Problem 3
Let Xi denote the gamblers net gain for the ith play. Then, X1 , X2 , , X100 are i.i.d.
random variables with: Xi = 1, with probabi
EE 805, Random Processes and Linear Systems
OSU, Winter 2010
Jan. 11, 2010
Due: Jan. 18, 2010
Problem Set 1
Problem 1
Let X be an exponentially distributed random variable with parameter , i.e., its pdf is fX (x) =
ex for x 0, and zero elsewhere.
(a) Let
Errata
Introduction to Probability, 2nd Edition
Dimitri P. Bertsekas and John N. Tsitsiklis
Athena Scientic, 2008
(last updated 12/25/11)
These are corrections to the rst and second printing of the book, and have been xed in the third printing.
Books from
3. Part (c)
probability of remaining time be greater than 0.5 hrs
0.7
0.6
0.5
0.4
0.3
part b
part c (i)
part c (ii)
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.2
1.4
1.6
1.8
2
Part (d)
1
0.9
0.8
0.7
FT(t)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.
ECE 804, Random Signal Analysis
OSU, Autumn 2009
Nov. 2, 2009
MIDTERM
Clearly print your name below.
This is a closed book, closed notes exam, 1 page cheat sheet is allowed.
The duration of the exam is 45 minutes. Questions carry dierent weights, which
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
August 23, 2012
Course Syllabus
Time and venue: Tu,Th 3:555:15 pm, Scott Lab E0125
Instructor: C. Emre Koksal, DL 712, [email protected]
Web page: Class material will be posted on Carmen
Oce Hou
A
B
A B
A B
AB
B A
Trial 1
0.305
Trial 2
0.3110 0.4230 0.6260 0.1080 0.2553 0.3473
Trial 3
0.2890 0.4090 0.5960 0.1020 0.2494 0.3529
Probability
0.3
0.3780 0.5860 0.0970 0.2566 0.3180
0.4
0.6
0.1
0.25
N=1000;
x=rand(N,1);
count1=0;
count2=0;
for n_index=1
ECE 804, Random Signal Analysis
OSU, Autumn 2009
Dec. 2, 2009
Solutions - Problem Set 8
Problem 1
(a)
fY (y) = fY (y | Z = 1) P (Z = 1) + fY (y | Z = 1) P (Z = 1)
1
y2
= exp
2
2
(b) By iterated expectations,
E [XY ] = E [E [XY | Z]
1
1
= E [XY | Z = 1] +