San Jose State University
Department of Electrical Engineering
Exam 1
EE 102, Section
Fall 2005
Closed Book, Closed Notes,
Instructions: There are four problems. Interpretation of questions will not be
given during the exam. If you are unsure about the me
EE 250: Probability, Random
Variables and Stochastic Processes
MIDTERM 1 SOLUTIONS
SJSU
FALL 2012
Problem 1
40
Problem 2
30
Problem 3
30
Total
100
Notes:
The exam is CLOSED book, notes, etc.
Show your work for full/partial credit
DO NOT FORGET TO WRITE
EE 250: Probability, Random
Variables and Stochastic Processes
MIDTERM 1 SOLUTIONS
SJSU
SPRING 2009
Problem 1
33
Problem 2
33
Problem 3
33
Attendance
1
Total
100
Notes:
The exam is CLOSED book, notes, etc.
Show your work for full/partial credit
You mig
Quiz 2.5] The probability that a call is a voice call is
probability of a data call is
P[ D ] 0.3
P[V ] 0.7
. The
. Voice calls cost 25 cents
each and data calls cost 40 cents each. Let C equal the cost of one
telephone call
Find.
1. PMF
PC c
.
2. The e
EE 250: Probability, Random
Variables and Stochastic Processes
MIDTERM 1
SJSU
FALL 2009 - Type A
CHOOSE THREE OF THE PROBLEMS!
Problem 1
33
Problem 2
33
Problem 3
33
Problem 4
33
Attendance
1
Total
100
Notes:
The exam is CLOSED book, notes, etc.
Show yo
FX,Y
EE 102
0
2y2 y4
2x
x4
(x, y) =
2y 2 y 4
1
x < 0 or y < 0
0<yx1
0 x y, 0 x 1
0 y 1, x 1
x 1, y 1
Problem Set 6
(15)
Solution
Problem 4.5.1 Solution
(a) The joint PDF (and the corresponding region of nonzero probability) are
Y
fX,Y (x, y) =
X
1
1/2 1 x
Probability, Statistics & Stochastic Processes
Stochastic processes
Lecture 20
Khosrow Ghadiri
Khosrow Ghadiri
Electrical Engineering Department
San Jose State University
1
Probability in terms of Set Theory
Sample point or outcome, s, of a random experi
San Jose State University
Department of Electrical Engineering
College of Engineering
Exam II
EE 250
Fall 2014
Closed Book, Closed Notes, No electronic devices
Instructions: There are four problems. Interpretation of questions
will not be given during the
EE 250
Final Exam
Final Exam Thursday Dec. 18, 2014 12:15-14:30
Dear EE 250 Students,
Final Exam entails 6 problems of 15 points each and one of 10 points.
Problem 1: two random Variables probability measures
Problem 2: Markov, chebyshev inequalities and
San Jose State University
Department of Electrical Engineering
Exam 1
Solution
EE 250
Fall 2013
Closed Book, Closed Notes,
Instructions: There are four problems. Interpretation of questions will not be
given during the exam. If you are unsure about the me
Moment Generating Function (MGF)
Motivation:
The expected values E X , E X 2 , E X 3 , , E X n are first through n-moments. The firstorder probability, E X X , the second-order probabilities such as variance
VAR X 2X E X 2 2X , and higher order probabilit
San Jose State University
Department of Electrical Engineering
College of Engineering
Exam II
Solution
EE 250
Fall 2014
Closed Book, Closed Notes, No electronic devices
Instructions: There are four problems. Interpretation of questions
will not be given d
San Jose State University
Department of Electrical Engineering
Final Exam
EE 250
Spring 2012
Closed Book, Closed Notes, No electronic devices
Instructions: There are five problems. Interpretation of questions will not be
given during the exam. If you are
San Jose State University
Department of Electrical Engineering
College of Engineering
Exam II
EE 250
Fall 2015
Solution
Closed Book, Closed Notes, No electronic devices
Instructions: There are four problems. Interpretation of questions
will not be given d
Conditional Probability Mass Function
Recall that
P[ A | B ]
Assume that
to be the observation of a particular value of random variable
A
A cfw_ X x
P[ AB]
P[ B ]
, then we can write the conditional probability mass function of
P[ X x, B]
P[ A | B ] P[
San Jose State University
Department of Electrical Engineering
Exam 1
EE 250
Fall 2013
Closed Book, Closed Notes,
Instructions: There are four problems. Interpretation of questions will not be
given during the exam. If you are unsure about the meaning of
Gaussian Random Vectors
Multivariate Gaussian distribution:
Probability model for
n
random variables with the property that the
marginal PDFs are all Gaussian.
A vector whose components are jointly Gaussian random variables is
said to be Gaussian rando
Lec 011
Unit impulse (Delta) function
Delta function plots
1
d x
0
For any continuous function
1
1
- x
2
2
o.w
d x dx 1
g x
,
g x x x0 dx g x0
1
Unit step function
1 x 0
u x
0 x 0
Using the definition of unit step function, we can write the CDF of
X
a
Continuous Random Variables
For the limits
x1 , x2
x1 , x2
x1
with
x1 x2
, there are four different interval cases.
x1
and
x2
x1 , x2 x | x1 x x2
: Interval defined as all numbers between
x1
and
x2
including
x1
and
x1 , x2 x | x1 x x2
: Interval d
Lec 01
Definitions
Axioms
Theorems
o Definitions establish the logic of probability theory.
o Axioms are facts that we accepts without proof.
o Theorems are consequences of definitions and axioms.
Set: Collection of things
Ex) How to define a set with
San Jos State University
College of Engineering
Electrical Engineering Department
EE250, Probability, Random Variables and Stochastic
Processes, Fall 2014
Course and Contact Information
Instructor:
John (JeongHee) Kim
Office Location:
ENG365
Telephone:
(4
2.4 Cumulative Distribution Function (CDF)
The cumulative distribution function of random variable
is
For any real number x, the CDF is the probability that the random variable
X is no larger than x.
For all
: Subscript corresponds to the name of the rand
Lec 12
Ex 3.24) Let
X
be a random variable with CDF
output of a clipping circuit with the characteristic
Express
g x
FY y
and
tells us that
fY y
Y
in terms of
FX x
and
FX x
. Let
Y
Y g ( X )
fx x
has only two possible values, 1 and 3. Thus
discret
Conditional probability
P A B
:
Probability of
Probability of
A
A
given
B
conditioned on
P A B
P B
P AB
P B
P AB
P A B
P AB P B P A B
Bays Theorem
P B | A
=
P A | B P B
P A
P AB
P B
P B
P A
P AB
=
=
P B
P B
P A
P AB
P A
B
Sequential experimen
Random Vectors
Generalization of the concept in ch. 4 to any # of random variables
Vector notation
Matrix notation
Probability model with
n
random variables
X 1 ,., X n
Multivariate Joint CDF
The joint CDF of
X 1 ,., X n
is
FX1 ,., X n x1 ,., xn P X 1
Lec 14
For two random variables X and Y
E X Y E X E Y
Joint PMF, PX ,Y x, y or PDF, f X ,Y x, y is not required for
E X Y E X E Y
However, the variance of X Y depends on the entire joint PMF, PDF or
joint CDF
Variance of sum of two random variables X an
Lec 15
Conditioning by a Random Variable
Conditional PMF
For any event Y y such that PY y 0 , the conditional PMF of X given
Y y is
PX |Y x | y P X x | Y y
P x, y P x | y P y P y | x P x
X |Y Y Y | X X
X ,Y
P x, y
PX |Y x | y X ,Y
PY y
Condition
Power Spectral Density
1
Power spectral density
Amount of power per unit (density) of frequency (spectral) as a function of the
frequency.
The power spectral density (PSD), which describes how the power of a signal or
time series is distributed with fre
Binomial Theorem
1
Ex]A system has six components. Each component has a failure probability , independent of
any other component. What is the probability that the operation is successful?
Write your answer in terms of .
If operation is successful, then
A
Lec 02
Set Algebra
Universal set
Set
Element
Probability
Sample space
Event
Outcome
Probability of Axioms
Indicates the probability of an event.
For any events
,
For any countable collection
of mutually exclusive events
1
Ex] Roll a fair die (6 si