yrd . doç . dr. didem kivanc tureli didemk@ieee [email protected]
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OKAN UNIVERSIT Y FACULTY OF ENGINEERING AND ARCHITECTURE MATH 256 Probability and Random Processes 01 Introduction Fall 2011. Yrd . Doç . Dr. Didem Kivanc Tureli [email protected] [email protected]. MATH 256: Probability and Random Processes. Instructor: Didem Kıvanç Türeli - PowerPoint PPT PresentationTRANSCRIPT
OKAN UNIVERSITYFACULTY OF ENGINEERING AND ARCHITECTURE
MATH 256 Probability and Random Processes
01 IntroductionFall 2011
Yrd. Doç. Dr. Didem Kivanc [email protected]
104/21/23 Lecture 1
MATH 256: Probability and Random Processes
• Instructor: Didem Kıvanç Türeli• [email protected]• [email protected]• Phone:?• Office: Engineering & Architecture Building 328• Class: Wednesday 11-12:00, 13:00-15:00 – Room D408• Course website:
http://personals.okan.edu.tr/didem.kivanc/courses/MATH256/
04/21/23 Lecture 1 2
MATH 256: Probability and Random Processes
• Textbook– A First Course in Probability (8th Edition) by Sheldon Ross, Eighth Edition,
Prentice Hall, ISBN: 013603313X / 0-13-603313-X– Probability, Random Variables and Random Signal Principles (4th Edition)
by Peyton Z. Peebles, Jr., ISBN: 0-07-366007-X / 978-0073660073
• Course Outline– Combinatorial Analysis, Axioms of Probability, Conditional Probability
and Independence, Random Variables, Jointly Distributed Random Variables, Properties of the Expectation Operator, Limit Theorems, Temporal Characteristics of Random Processes, Poisson Processes and Markov Chains.
• Prerequisites– Calculus I,II,III,IV
04/21/23 Lecture 1 3
EE 403 Grading Policy• Homework: Homework will be mandatory.• Exams: 2 Midterm exams and 1 Final Exam• Attendance: 70% is required.• Quizzes are randomly given at any time of the class.• Grading: The final course grade will be based on
homework, quizzes, midterm exams, and final exam.• Grading Formula:
0.02xQuiz+0.08xHomework+0.4xMidterms+0.5xFinal
04/21/23 Lecture 1 4
Academic Dishonesty
• Any violation of academic integrity will receive academic and possibly disciplinary sanctions. The sanctions include the possible awarding of a F grade.
• Cheating• Copying on a test• Plagiarism• Acts of aiding or abetting• Submitting previous work• Tampering with work• Altering exams
04/21/23 Lecture 1 5
Course Objectives• Upon completing the course, students will be able to analyze
systems which involve random variables.
04/21/23 Lecture 1 6
Course Outline (1/2)
04/21/23 Lecture 1 7
Course Plan
Week 1.(Sep. 21st) Combinatorial AnalysisWeek 2.(Sep. 28th) Axioms of ProbabilityWeek 3.(Oct. 5rd) Conditional Probability and Independence
Week 4.(Oct. 12th) Random Variables
Week 5.(Oct. 19th) Continuous Random Variables Week 6.(Oct. 26th) Midterm 1Week 7.(Nov. 2nd) Jointly Distributed Random Variables
Course Outline (2/2)
04/21/23 Lecture 1 8
Week 8.(Nov. 9th) No Class (Kurban Bayramı)
Week 9.(Nov. 16th) Properties of Expectation
Week 10.(Nov. 23rd) Limit Theorems
Week 11.(Nov. 30th) Midterm Exam-II
Week 12.(Dec. 7th) Temporal Characteristics of Random Processes
Week 13.(Dec. 14th) Temporal Characteristics of Random Processes
Week 14.(Dec. 21st) Poisson Processes and Markov Chains
Final Exam 02/01 - 13/01 2012
Example of how we use probability• Consider a communication system with multiple antennas.
We need at least one antenna working correctly for the system to work.
• Any antenna can fail with probability ½. • What is the probability that the system will fail? (i.e. none of
the antennas will work)
04/21/23 Lecture 1 9
Look at all the possibilities …• Count the number of ways things can occur.
• The mathematical theory of counting is called combinatorial analysis.
04/21/23 Lecture 1 10
Antenna 1 Antenna 2 Antenna 3 Antenna 4 Antenna 5
1 1 1 1 1
1 1 1 1 0
1 1 1 0 1
1 1 1 0 0
1 1 0 1 1
1 1 0 1 0
1 1 0 0 1
The basic principle of counting• Two experiments are performed. • If experiment 1 can result in any one of m possible outcomes,
and • experiment 2 can result in any one of n possible outcomes,
then • together there are mxn possible outcomes of the two
experiments.
04/21/23 Lecture 1 11
The generalize basic principle of counting• r experiments– n1 possible outcomes of first experiment
– For each of n1 possible outcomes, n2 possible outcomes of 2nd experiment,
– For each of n2 possible outcomes, n3 possible outcomes of 3rd experiment,
– …• Then there are a total of n1 x n2 x n3 x … x nr possible
outcomes of the r experiments.
04/21/23 Lecture 1 12
Examples• How many 7 place license plates are possible if the first 3
places are to be occupied by letters and the final 4 by numbers?
• (assume 26 letter alphabet)• 26 x 26 x 26 x 10 x 10 x 10 x 10
• What if you don’t allow repetition of letters or numbers?
04/21/23 Lecture 1 13
Permutations• How many different ordered arrangements of the letters a,b,c
are possible?• abc, bac, acb, cab, bca, cba• 3 x 2 x 1 = 6 permutations• Number of permutations of n objects:– n! = n x (n-1) x (n-2) x … x 1
• How many arrangements using the letters PEPPER?
04/21/23 Lecture 1 14
• Each letter is different: P1, E1, P2, P3, E2, R1
04/21/23 Lecture 1 15
P1 E1 P2 P3 E2 R1 P1 E2 P2 P3 E1 R1 E1 P1 P2 P3 E2 R1 E2 P1 P2 P3 E1 R1
P2 E1 P1 P3 E2 R1 P2 E2 P1 P3 E1 R1 E1 P2 P1 P3 E2 R1 E2 P2 P1 P3 E1 R1
P1 E1 P3 P2 E2 R1 P1 E2 P3 P2 E1 R1 E1 P1 P3 P2 E2 R1 E2 P1 P3 P2 E1 R1
P3 E1 P1 P2 E2 R1 P3 E2 P1 P2 E1 R1 E1 P3 P1 P2 E2 R1 E2 P3 P1 P2 E1 R1
P2 E1 P3 P1 E2 R1 P2 E2 P3 P1 E1 R1 E1 P2 P3 P1 E2 R1 E2 P2 P3 P1 E1 R1
P3 E1 P2 P1 E2 R1 P3 E2 P2 P1 E1 R1 E1 P3 P2 P1 E2 R1 E2 P3 P2 P1 E1 R1
6!Number of permutations
3!2!1!
Number of RsNumber of PsNumber of Es
Number of permutations in general…• If there are n objects, of which n1 are alike, n2 are alike, n3 are
alike, …, nk are alike, then
04/21/23 Lecture 1 16
1 2
!Number of permutations
! ! !k
n
n n n
Combinations• Determine number of different groups of r objects that could
be formed from a total of n objects. • This time there is no order to the objects: abc is the same as
cab.• Example: How many ways are there to choose 3 letters from
the English alphabet (26 letters)?– {a,b,c}, {a,b,d}, {a,b,e}, …– {a,c,b}, {a,c,d}, {a,c,e},…– {a,b,c} is the same as {a,c,b}– How many groups are the same? Or to ask this differently– How many permutations of 3 letters are there? 3!
04/21/23 Lecture 1 17
Combinations• Number of different groups of r items that could be formed
from a set of n items
04/21/23 Lecture 1 18
( 1) ( 1) ( 1) !
! ( )! !
n n n n r n
r n r r
Define for !
( , )( )! !
rnn
C n rr n r r
n
• DEFINITION: The number of possible combinations of n objects taken r at a time
• Useful identity:
04/21/23 Lecture 1 19
1 11
n n nr r r
• THEOREM: The Binomial Theorem
0
( )n
n k n k
k
nx y x y
k
Multinomial Coefficients• A set of n distinct items is to be divided into r distinct groups
of respective sizes n1, n2, …, nr, where
• How many different divisions are possible?• There are – C(n, n1) possible choices for the first group
– C(n-n1, n2) possible choices for the second group
– C(n - n1 - n2, n3) possible choices for the third group, …
04/21/23 Lecture 1 20
0
.r
ii
n n
04/21/23 Lecture 1 21
1 1 2 1
1 2
1 1 2 1
1 1 1 2 2
1 2
! !!
! ! ! ! 0! !!
! ! !
r
r
r
r
r
n n n n n nnn n n
n n n n n nn
n n n n n n n nn
n n n
Multinomial Theorem
04/21/23 Lecture 1 22
1 2
1 21 2 1 2
Define for !
( , , , , ), , , ! ! !
r
rr r
n n nnn
C n n n nn n n n n n
n
• DEFINITION:
• THEOREM: The Multinomial Theorem
1 2
1
1
1 2 1 21 2( , , ):
( ), , ,
r
r
r
n n nnr r
rn nn n n
nx x x x x x
n n n
Distribution of Balls in Urns• x1+
04/21/23 Lecture 1 23