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
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
Problem Set 6
Random Variations & Stochastic Systems
PROBLEM 1:
The number of calls coming per minute into a computer repair center is Poisson random variable with
mean 3.
(a) Find the probability that no calls come in a given 1 minute period.
(b) Assume
Lecture 1
Set Theory as
a Language of
Probability
1
Random Signals & Stochastic Processes
Set Theory
A set is a collection of objects.
The objects in a set are referred to as the elements, or
members of the set.
A set is denoted by a capital letter whi
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EE 210 Signal, Sequence and Digital and Analog System 1
PROBLEM 1:
Given the probability mass function
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x
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0
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Lecture 14
Markov
Inequality
Random Signals & Stochastic Processes
1
Markov inequality
Rational: A known mean and variance of a random variable, X do
not provide enough information to determine the CDF and PDF.
However, the mean and variance of a random
Problem Set 8
Random Variations & Stochastic Systems
PROBLEM 1:
Given the joint probability density function
Ax
0
3 x y 20
f XY x, y
0x4
otherwise
a) Find P Y 5 X 0 X 2 .
b) Find f X x and fY y .
c) Directly find f X x Y 9 and fY y X 2
d) Find f X x Y y
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1.
Table of Contents
Probability Theory . 2
2.
The Relative Frequency Definitions of Probabilities . 3
3.
Sequential experiment . 4
Khosrow Ghadiri 2013
Probability Theory
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robaility is the analysis of random phenomena. The centeral objects of
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San Jose State University
Department of Electrical Engineering
College of Engineering
Exam 1
EE 250
Fall 2016
Closed Book, Closed Notes, No electronic devices
Instructions:
There are four problems. They are weighted as shown. Interpretation of
questions
S O L U T Io N 0 6 2
OB L E M
h r e e ty p e s
h ig h p r i o r it y
1 1 1 B R A N D O M
S IG N A L S & S T O C H A S T I C P R O C E S S E S
61
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RANDOM SIGNALS & STOCHASTIC PROCESSES
PROBLEM 1:
a) Prove A B B
b) Prove A
B B
A
A
1
PROBLEM 001
SOLUTION 001
RANDOM SIGNALS & STOCHASTIC PROCESSES
PROBLEM 1:
a) Prove A
B B
A
b) Prove A
B B
A
SOLUTION 1:
1a
x A x B x B x A
x A B
x A x B
x B A
x B x
15 1
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Problem Set 7
Random Variations & Stochastic Systems
PROBLEM 1:
A random variable X and Y has the joint CDF
(1 e x )(1 e y ) x 0, y 0
FX ,Y x, y
otherwise
0
a) Find the P X 2, Y 3
b) Find the marginal CDF, FX x .
c) Find he marginal CDF, FY y .
PROBLEM 2
Probability, Statistics & Stochastic Processes
Markov Chains
Lecture 25
Khosrow Ghadiri
Khosrow Ghadiri
Electrical Engineering Department
San Jose State University
1
Markov Chains
Future is independent of the past given the present
Andrei Markov (185619
PROBLEM :
Two competitors Jack and John play a game. 4 coins are tossed, Jack wins if exactly 2 heads
are obtained and John wins if exactly 3 heads or 3 tails occur. Otherwise the coins are tossed
again. Find:
SOLUTION:
a) Probability [Jack wins].
4!
6 3
San Jose State University
Department of Electrical Engineering
Exam I
EE 250
Fall 2015
Closed Book, Closed Notes, No electronic devices
Instructions: There are four problems. Interpretation of questions
will not be given during the exam. If you are unsure
Lecture 16
Markovs
Inequality 2
Random Variations & Stochastic Systems
1
Bounds summary
Sum bound: A random variable, X is defined as anoccurrence
of the number of events E1 , E2 , En , then:
E X P E1 P E2 P En
Expectation linearity : The expectation o
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5.1 Discrete Probability Distribution Models
A discretevalue random probability distribution has a set of distinct sample point
separated by values that cannot occur together with the likelihood of occurrence
SAN JOSE STATE UNIVERSITY
COLLEGE OF ENGINEERING
DEPARTMENT OF ELECTRICAL ENGINEERING
EE 250, Probability, Random Variables and Stochastic
Processes, Section 03 Spring 2017
Instructor:
Khosrow Ghadiri
Office Location:
ENGR 371
Telephone:
(408) 9243916
Fa
San Jose State University
Department of Electrical Engineering
College of Engineering
Exam 2
EE 250
Fall 2016
Closed Book, Closed Notes, No electronic devices
Instructions:
There are four problems. They are weighted as shown. Interpretation of
questions
Lecture 1
Conditional
Probability
Random Variations & Stochastic Systems
1
Probability in Terms of Set Theory
Conditional Probability:
s1 , s2 ,
P E1 E2
P E1 E2
P E2
Sample Space S
E1
E1
E2
Random Variations & Stochastic Systems
E2
2
Conditional prob
1.0 INTRODUCTION
Perhaps one of the more important application areas of digital signal processing (DSP) is the power spectral estimation
of periodic and random signals. Speech recognition problems use spectrum analysis as a preliminary measurement
to perf
Lecture 25
Markovs
Chain
1
Markov Chain
Future is independent of the past given the present
Andrei Markov (18561922)
2
Markov Chain and Process
Markov Chain:
Stochastic processes S n and S t are a Markov chain and
Markov Process if the future of the pr
Lecture 16
Markov
Inequality 2
1
Bounds summary
Sum bound: A random variable, X is defined as the number
of an occurrence of the number of events E1 , E2 , En , then:
E X P E1 P E2 P En
Expectation linearity : The expectation of sum of the events
E1 ,