Fourth Problem Assignment
EECS 401
Problem 1 Joe and Helen each know that the a priori probability that her mother will
be home on any given night is 0.6. However, Helen can determine her mothers plan
for the night at 6 P.M., and then, at 6:15 P.M., she h
First Problem Assignment
EECS 401
Problem 1 Fully explain your answers to the following questions.
(a) If events A and B are mutually exclusive and collectively exhaustive, are Ac and
Bc mutually exclusive?
(b) If events A and B are mutually exclusive but
Second Problem Assignment
EECS 401
Problem 1 Bo and Ci are the only two people who will enter the Rover Dog Fod jingle
contest. Only one entry is allowed per contestant, and the judge (Rover) will declare
the one winner as soon as he receivers a suitably
Sixth Problem Assignment
EECS 401
Problem 1 An ambulance travels back and forth, at a constant speed, along a road of
length L. At a certain moment of time an accident occurs at a point uniformly distributed on the road. (That is, its distance from one of
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Electrical and Computer Engineering
ECE 139
Probability and Statistics
Spring 2011
Homework Assignment #8
(Due on Thursday 6/2/2011 by 8 pm in the Homework Box)
A. Solve the following problems in the t
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Electrical and Computer Engineering
ECE 139
Probability and Statistics
Spring 2011
Homework Assignment #7
(Due on Tuesday 5/24/2011 by 8 pm in the Homework Box)
Assignment: Solve the following problems
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Electrical and Computer Engineering
ECE 139
Probability and Statistics
Spring 2011
Homework Assignment #6 Part 2
(Due on Tuesday 5/17/2011 by 8 pm in the Homework Box)
The following are two ramp-up and
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Electrical and Computer Engineering
ECE 139
Probability and Statistics
Spring 2011
Homework Assignment #3
(Due on Tuesday 4/19/2011 by 8 pm in the Homework Box)
A. Solve the following problems in the t
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Electrical and Computer Engineering
ECE 139
Probability and Statistics
Spring 2011
Homework Assignment #2
(Due on Tuesday 4/12/2011 by 8 pm in the Homework Box)
Solve the following problems in the text
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Electrical and Computer Engineering
ECE 139
Probability and Statistics
Spring 2011
Homework Assignment #1
(Due on Tuesday 4/5/2011 by 8 pm in the Homework Box)
Please indicate whether you are Sophomore
(c) To nd the correlation, we evaluate the product XY over all values of X and Y . Specically,
x
4
rX,Y = E [XY ] =
xyPX,Y (x, y )
(8)
x=1 y =1
12
3
4
4
6
8
9
12 16
+
+
+
+
+
+
+
4 8 12 16 8 12 16 12 16 16
=5
=
(9)
(10)
(d) The covariance of X and Y is
Co
Problem Solutions Chapter 4
Problem 4.1.1 Solution
(a) The probability P [X 2, Y 3] can be found be evaluating the joint CDF FX,Y (x, y ) at
x = 2 and y = 3. This yields
P [X 2, Y 3] = FX,Y (2, 3) = (1 e2 )(1 e3 )
(1)
(b) To nd the marginal CDF of X, FX (
where u() denotes the unit step function. Since P [G] = 0.7, we can write
FD (y ) = P [G] P [D y |G] + P [Gc ] P [D y |Gc ]
0.3u(y )
y < 60
0.3 + 0.7(1 e(y60)/10 ) y 60
=
(4)
(5)
Another way to write this CDF is
FD (y ) = 0.3u(y ) + 0.7u(y 60)(1 e(y60)/10
3
a=6
a=3
a=0
a=3
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
Problem 3.3.1 Solution
fX (x) =
1/4 1 x 3
0
otherwise
(1)
We recognize that X is a uniform random variable from [-1,3].
(a) E [X ] = 1 and Var[X ] =
(3+1)2
12
= 4/3.
(b) The new random variable Y is
Useful Formulas and Denitions
1. Conditional Probability
P [A|B ] =
P [AB ]
P [B ]
2. Bayes Theorem
P [B |A] =
P [A|B ]P [B ]
P [ A]
3. Independence
P [AB ] = P [A]P [B ]
4. n choose k
n
n!
=
k !(n k )!
k
5. CDF
FX (x) = P [X x]
6. CDF and PDF
x
FX (x) =
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Electrical and Computer Engineering
ECE 139
Spring 2011
Instructor: K. Rose
Final Examination
June 2011
Instructions: The exam is open book and open notes. It consists of four problems
which are not of
ECE 139 HW 4 Solutions
Problem 2.2.4:
Part a:
ccc
1
248
7
c 1
8
8
c
7
Part b:
c 18 2
4 47 7
P X 4 PX 4
Part c:
P X 4 P X 2 PX 2
c 18 4
2 2 7 7
Part d:
P 3 X 9 P X 4 P X 8 PX 4 PX 8
213
777
Problem 2.2.6:
P X 1 p
P X 2 p 1 p
P X 3 p 1 p
2
0.95 P
ECE 139 HW 3 Solutions
Part A:
Problem 1.6.4:
Part a:
P A B 0 (since A and B are disjoint)
P B P A B P A
P A B c P A
5321
8884
3
8
P A Bc P Bc 1 P B 1
13
44
Part b:
P AB 0 P A P B
Therefore, NOT independent
Part c:
P C D P C P D
11
P D
32
2
P D
3
1
ECE 139 HW 2 Solutions
Problem 1.4.1:
P H 0 P H 0 L P H 0 B 0.1 0.4 0.5
P B P BH 0 P BH1 P BH 2 0.4 0.1 0.1 0.6
P L H 2 P L P H 2 P L H 2 0.4 0.3 0.2 0.5
Problem 1.4.2:
P B1 1
11
P B2 1
0.171
1 0.142
31
P B3 1
21
1 0.118
P B4 1
P B5 1
2
41
1 0.09
ECE 139 HW 1 Solutions
Problem #1:
A = cfw_3, 4, 5, 6
B = cfw_1, 3, 5
a.
b.
c.
d.
e.
S = cfw_1, 2, 3, 4, 5, 6
A B = cfw_1, 3, 4, 5, 6
A B = cfw_3, 5
A B = cfw_4, 6
Ac Bc = cfw_2
Problem #2:
a. No. If A B and A C are equal, it does not mean that B and C ar
Seventh Problem Assignment
EECS 401
Problem 1 Random variables X and Y are independent and are described by the probability density functions fX (x) and fY (y)
fX (x)
0
fY ( y )
x (hours)
1
0
y (hours)
1
Stations A and B are connected by two parallel mess
Fifth Problem Assignment
EECS 401
Problem 1 (24 points) Discrete random variable X is described by the PMF
pX (x) =
K
0,
x
12 ,
if x = 0, 1, 2
for all other values of x
Let D1 , D2 , . . . , DN represent N successive independent experimental values of ran
Third Problem Assignment
EECS 401
Problem 1 Oscar has lost his dog in either forest A (with a priori probability 0.4) or
in forest B (with a priori probability 0.6). If the dog is alive and not found by the Nth
day of the search, it will die that evening
First Problem Assignment
EECS 401
Problem 1 Fully explain your answers to the following questions.
(a) If events A and B are mutually exclusive and collectively exhaustive, are Ac and
Bc mutually exclusive?
Solution Ac Bc = (A B)c = c = . Thus the events