ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Sep. 4, 2012
Solutions - Problem Set 1
Problem 1
(a) F must contain all intersections, unions, and complements the two sets, so
cfw_2 , cfw_1 , 3 , 4 , cfw_1 , 2 , 4 , cfw_3
are elements. Now, i
Solutions - Problem Set 2
Problem 1
(a) Since each element in the sequence takes on 3 possible values, the number of potential
sequences is 3n .
(b) In this part, changing the location of zeros, ones and twos does not change the sequence. This leads us to
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Sep. 13, 2012
Solutions - Problem Set 2
Problem 1
(a) Since each element in the sequence takes on 3 possible values, the number of potential
sequences is 3n .
(b) In this part, changing the locat
ECE 804, Random Signal Analysis
OSU, Autumn 2010
Oct. 4, 2010
Due: Oct. 11
Problem Set 2
Problem 1
Let X1 , . . . , Xn be a sequence of numbers, where each number takes on the value 0, 1 or
2 with probability 1 , 1 and 1 respectively.
24
4
(a) What is the
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Sep. 13, 2012
Due: Sep. 25, 2012
Problem Set 3
Problem 1
We consider a random selection of coins, where the probability of heads, C for the coins
is a random variable whose pdf is fC (c) = k c fo
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Sep. 25, 2012
Solutions - Problem Set 3
Problem 1
(a)
1=
1
fC (c) dc =
0
1
c2
kc dc = k
2
Thus, fC (c) =
=
0
k
2
=
k=2
2c, 0 c 1
0, otherwise
(b)
1
P (H ) =
P (H |c) fC (c) dc =
c 2c dc = k
0
2 c
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Oct. 2, 2012
Solutions - Problem Set 4
Problem 1
(a) K0 =
1
100
and K1 =
1
,
5050
since
100
1=
K1 x = K1
x=1
100 101
.
2
(b) First,
P (X = x) = P (X = x | H0 ) P (H0 ) + P (X = x | H1 ) P (H1 ) =
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Nov. 15, 2012
Due: Nov. 29, 2012
Problem Set 8
Problem 1
Let X be an exponentially distributed random variable with parameter . This problem is about
estimating X , based on the observation Y , w
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Oct. 18, 2012
Due: Nov. 1, 2012
Problem Set 7
Problem 1
At a party n people put their hats in the center of a room, where the hats are mixed
together. Each person then randomly selects one with e
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Oct. 11, 2012
Due: Oct. 18, 2012
Problem Set 6
Problem 1
Consider the function (for c > 0)
f (x, y ) =
2 |x|, c x c, 1 y 1
0,
otherwise
(a) Find the constant c for which f (x, y ) is the joint pd
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Oct. 4, 2012
Due: Oct. 11, 2012
Problem Set 5
Problem 1
We want to obtain the mold content per volume, m, of the water in the Dreese building,
with an error that, with 95 % probability, is less t
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Sep. 25, 2012
Due: Oct. 2, 2012
Problem Set 4
Problem 1
Suppose the conditional probabilities for a students score X cfw_1, 2, . . . , 100 on a test
are as follows:
Under the hypothesis H0 (the
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Sep. 4, 2012
Due: Sep. 13, 2012
Problem Set 2
Problem 1
Let X1 , . . . , Xn be a sequence of numbers, where each number takes on the value 0, 1 or
1
2 with probability 1 , 1 and 4 respectively.
2
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Aug. 28, 2012
Due: Sep. 4, 2012
Problem Set 1
Problem 1
For the set = cfw_1, 2, 3, 4,
(a) Find the minimal eld that contains the subsets cfw_1, 2 and cfw_2, 4.
(b) Find the minimal eld that conta
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Oct. 11, 2012
Solutions - Problem Set 5
Problem 1
(a)
1
E [Mn ] = E
n
n
Xi
i=1
1
=
n
n
E [Xi ]
i=1
=m
and
1
n
var (Mn ) = var
=
1
n2
n
Xi
=
i=1
1
var
n2
n
Xi
i=1
n
var (m + Ni )
i=1
1
4
= 2 n 22
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Oct. 18, 2012
Solutions - Problem Set 6
Problem 1
(a)
c
1=
fX,Y (x, y ) dx dy = 2
c
= 4 2c
c
(2 |x|) dx = 2 2
c2
2
8
0
(2 x) dx
64 8
= 2 7/2
4
But if c > 2 fX,Y (x, y ) < 0 for some values of x,
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Nov. 1, 2012
Solutions - Problem Set 7
Problem 1
(a) As the ith person is equally likely to select any of the n hats, it follows that P (Xi = 1) =
1/n, and so
1
,
n
1
n1
1
.
var (Xi ) = 2 =
nn
n2
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, koksal@ece.osu.edu
Web page: Class material will be posted on Carmen
Oce Hou
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Oct. 4, 2012
Solutions - Midterm 1
Problem 1
(a) False. Since the balls are drawn without replacement, RR cannot be a valid experimental outcome. Every other pair is in , therefore | = 8.
(b) Fal
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Oct. 4, 2012
MIDTERM 1
NAME:
The duration of the exam is 80 minutes. The exam consists of 3 questions
with multiple parts each. Questions carry dierent weights, which are noted
next to the questi
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Nov. 6, 2012
Solutions - Midterm 2
Problem 1
For each of the following statements, answer True if the statement is always true, and
answer False otherwise. For each answer provide a very brief ar
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Nov. 6, 2012
Midterm 2
NAME (3 points):
The duration of the exam is 80 minutes. The exam consists of 3 questions
with multiple parts each. Questions carry dierent weights, which are noted
next to
ECE 804, Random Signal Analysis
OSU, Autumn 2010
Oct. 11, 2010
Due: Oct. 20
Problem Set 3
Problem 1
We consider a random selection of coins, where the probability of heads, C for the coins is a random
variable whose pdf is fC (c) = k c for 0 c 1 and zero
ECE 804, Random Signal Analysis
OSU, Autumn 2010
Sep. 24, 2010
Due: Oct. 4
Problem Set 1
Problem 1
For the set = cfw_1, 2, 3, 4,
(a) Find the minimal eld that contains the subsets cfw_1, 2 and cfw_2, 4.
(b) Find the minimal eld that contains the subsets c