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School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e 11 Denition 1.4 Axioms of Probability A probability measure P[] is a function that maps events in the sample space to real numbers such that Axiom 1 For any event A, P[A] 0. Axiom 2 P[S] = 1. Axiom 3 For any countable collection A1, A2,
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e Section 2.1 Tree Diagrams 35 Yates & Goodman 3e Problem 2.1.5 57 Suppose that for the general population, 1 in 5000 people carries the human immunodeciency virus (HIV). A test for the presence of HIV yields either a positive (+) or nega
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e 20 Theorem 1.8 For a partition B = cfw_B1, B2, . . . and any event A in the sample space, let Ci = A \ Bi. For i 6= j, the events Ci and Cj are mutually exclusive and A = C1 [ C2 [ . Yates & Goodman 3e Theorem 1.9 For any event A, and p
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
ECE#226# Course#Informa0on# Sakai# Everything#is#on#the#Sakai#site# Syllabus# Homework#Problem#Sets# Homework#Solu0ons# Student#Solu0on#Manual# Annotated#Lecture#Slides# Ask#Public#Ques0ons#in#the#sakai#chatroom# If#you#need#a#quick#reply,#send#me
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
PROBABILITY AND STOCHASTIC PROCESSES A FRIENDLY INTRODUCTION FOR ELECTRICAL AND COMPUTER ENGINEERS Third Edition Chapter 7 Viewgraphs Roy D. Yates & David J. Goodman Yates & Goodman 3e Section 7.1 Conditioning a Random Variable by an Event 242 Yates & Goo
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
PROBABILITY AND STOCHASTIC PROCESSES A FRIENDLY INTRODUCTION FOR ELECTRICAL AND COMPUTER ENGINEERS Third Edition Chapter 3 Viewgraphs Roy D. Yates & David J. Goodman Yates & Goodman 3e 62 Discrete Random Variables In this chapter and for most of the rema
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e 11 Denition 1.4 Axioms of Probability A probability measure P[] is a function that maps events in the sample space to real numbers such that Axiom 1 For any event A, P[A] 0. Axiom 2 P[S] = 1. Axiom 3 For any countable collection A1, A2,
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e Section 2.1 Tree Diagrams 35 Yates & Goodman 3e Problem 2.1.5 57 Suppose that for the general population, 1 in 5000 people carries the human immunodeciency virus (HIV). A test for the presence of HIV yields either a positive (+) or nega
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e 20 Theorem 1.8 For a partition B = cfw_B1, B2, . . . and any event A in the sample space, let Ci = A \ Bi. For i 6= j, the events Ci and Cj are mutually exclusive and A = C1 [ C2 [ . Yates & Goodman 3e Theorem 1.9 For any event A, and p
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
ECE#226# Course#Informa0on# Sakai# Everything#is#on#the#Sakai#site# Syllabus# Homework#Problem#Sets# Homework#Solu0ons# Student#Solu0on#Manual# Annotated#Lecture#Slides# Ask#Public#Ques0ons#in#the#sakai#chatroom# If#you#need#a#quick#reply,#send#me
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
PROBABILITY AND STOCHASTIC PROCESSES A FRIENDLY INTRODUCTION FOR ELECTRICAL AND COMPUTER ENGINEERS Third Edition Chapter 7 Viewgraphs Roy D. Yates & David J. Goodman Yates & Goodman 3e Section 7.1 Conditioning a Random Variable by an Event 242 Yates & Goo
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
PROBABILITY AND STOCHASTIC PROCESSES A FRIENDLY INTRODUCTION FOR ELECTRICAL AND COMPUTER ENGINEERS Third Edition Chapter 3 Viewgraphs Roy D. Yates & David J. Goodman Yates & Goodman 3e 62 Discrete Random Variables In this chapter and for most of the rema
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: NetID: 332:226 Signature: Probability and Stochastic Processes February 28, 2013 Examination 1 You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted.
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
332:226 Probability and Stochastic Processes April 18, 2013 SOLUTION (VERSION 1) (The solution for version 2 starts on page 7.) Examination 2 You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; nei
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: 332:226 NetID: Signature: Probability and Stochastic Processes February 26, 2015 Examination 1 You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted.
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
332:226 Probability and Stochastic Processes February 28, 2013 SOLUTION Examination 1A You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted. Make sure tha
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
 ,.,. A .i 332:226 Signature: _ _ _ _ _ _ _ Prob?bility and Stochastic Processes February 26, 2015 Examination 1 You have 110 minutes to answer the following questions in the:notebooks provided . This is. a c'losed book exam; neither notes nor calculator
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: NetID: 332:226 Signature: Probability and Stochastic Processes April 18, 2013 Examination 2 You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted. Ma
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e 11 Denition 1.4 Axioms of Probability A probability measure P[] is a function that maps events in the sample space to real numbers such that Axiom 1 For any event A, P[A] 0. Axiom 2 P[S] = 1. Axiom 3 For any countable collection A1, A2,
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e Section 2.1 Tree Diagrams 35 Yates & Goodman 3e Problem 2.1.5 57 Suppose that for the general population, 1 in 5000 people carries the human immunodeciency virus (HIV). A test for the presence of HIV yields either a positive (+) or nega
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e 20 Theorem 1.8 For a partition B = cfw_B1, B2, . . . and any event A in the sample space, let Ci = A \ Bi. For i 6= j, the events Ci and Cj are mutually exclusive and A = C1 [ C2 [ . Yates & Goodman 3e Theorem 1.9 For any event A, and p
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
ECE#226# Course#Informa0on# Sakai# Everything#is#on#the#Sakai#site# Syllabus# Homework#Problem#Sets# Homework#Solu0ons# Student#Solu0on#Manual# Annotated#Lecture#Slides# Ask#Public#Ques0ons#in#the#sakai#chatroom# If#you#need#a#quick#reply,#send#me
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
PROBABILITY AND STOCHASTIC PROCESSES A FRIENDLY INTRODUCTION FOR ELECTRICAL AND COMPUTER ENGINEERS Third Edition Chapter 7 Viewgraphs Roy D. Yates & David J. Goodman Yates & Goodman 3e Section 7.1 Conditioning a Random Variable by an Event 242 Yates & Goo
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
PROBABILITY AND STOCHASTIC PROCESSES A FRIENDLY INTRODUCTION FOR ELECTRICAL AND COMPUTER ENGINEERS Third Edition Chapter 3 Viewgraphs Roy D. Yates & David J. Goodman Yates & Goodman 3e 62 Discrete Random Variables In this chapter and for most of the rema
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e 66 Probability Mass Function Denition 3.3 (PMF) The probability mass function (PMF) of the discrete random variable X is PX (x) = P [X = x] Yates & Goodman 3e 68 Families of Random Variables In practical applications, certain families
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: NetID: 332:226 Signature: Probability and Stochastic Processes February 28, 2013 Examination 1 You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted.
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
332:226 Probability and Stochastic Processes April 18, 2013 SOLUTION (VERSION 1) (The solution for version 2 starts on page 7.) Examination 2 You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; nei
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: 332:226 NetID: Signature: Probability and Stochastic Processes February 26, 2015 Examination 1 You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted.
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
332:226 Probability and Stochastic Processes February 28, 2013 SOLUTION Examination 1A You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted. Make sure tha
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
 ,.,. A .i 332:226 Signature: _ _ _ _ _ _ _ Prob?bility and Stochastic Processes February 26, 2015 Examination 1 You have 110 minutes to answer the following questions in the:notebooks provided . This is. a c'losed book exam; neither notes nor calculator
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: NetID: 332:226 Signature: Probability and Stochastic Processes April 18, 2013 Examination 2 You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted. Ma
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: NetID: 332:226 Signature: Probability and Stochastic Processes April 15, 2015 Examination 2A Solution You have 110 minutes to answer the following questions on this examination paper. This is a closed book exam; neither notes nor calculators are per
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: 332:226 NetID: Signature: Probability and Stochastic Processes Final Examination May 9, 2013 MAKE SURE TO READ BOTH SIDES OF THIS EXAM PAPER. You have 180 minutes to answer the following questions in the notebooks provided. Make sure that you have i
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: NetID: 332:226 Probability and Stochastic Processes Calculus Prerequisite Quiz Quiz 0 You have 20 minutes to answer the following questions on this piece of paper. Calculators, textbooks, cheat sheets and other study aids are NOT permitted. Please
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: 332:226 NetID: Probability and Stochastic Processes Jan 2630, 2009 Quiz 1 You have 20 minutes to answer the following questions on this piece of paper. Calculators, textbooks, cheat sheets and other study aids are NOT permitted. Please make sure th
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
332:226 Probability and Stochastic Processes March 12, 2009 Examination 1 You have 110 minutes to answer the following four questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted. Make sure that you ha
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
226 Midterm 1 Feb 28, 2012 8:00pm Lucy Stone Hall Auditorium Livingston
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Announcements Midterm 1: 2/28, 8:0010:00pm, Lucy Stone Hall Auditorium (LSH AUD) Covers material through the 2/20 lecture
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Monday, January 28, 2013 12:06 PM Lectture Page 1 Lectture Page 2 Lectture Page 3 Lectture Page 4 Lectture Page 5 Lectture Page 6 Lectture Page 7 Lectture Page 8 Lectture Page 9 Lectture Page 10 Lectture Page 11 Lectture Page 12 Lectture Page 13 Lectture
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Announcements Midterm 1: 2/28, 8:0010:00pm, Lucy Stone Hall Auditorium (LSH AUD) Covers material through the 2/20 lecture
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
kPermutation: ordered sequence of k objects Permutations vs Combinations Permutation: order of selection matters Choose a sequence of 3 cards out of 52 Combination: order of selection does not matter Choose 3 cards out of 52 n choose kkcombinati
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
A B C A B A B C D C CD D AB D A C D C D A B AB A A B C D C CD D C D C D Hi Ri i i Hi hidden I Ri reveals i Hi hidden I Ri reveals i
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Definition 2.4 Probability Mass Function (PMF) The probability mass function (PMF) of the discrete random variable X is PX(X)=P[X=X] Definition 2.5 Bernouiii (p) Random Variable X is a Bernoulli (p) random variable if the PMF of X has the form 1 p x = 0 P
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
14:332:226 Probability and Stochastic Processes Probability and Random Processes Random Process: Observe a random signal/waveform Examples: Traffic at a webserver Calls at a telephone switch Noise in a receiver Unemployed people Species population Y
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Oneoftheseisnotliketheothers. N=2400 random variables
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
X1,Xn iid: X1,Xn iid, Bernoulli (p) W= X1+Xn W=X+Y (X,Y independent) PDF of W? MGF of W? W=X1+Xn (X1 Xn independent) PDF of W? MGF of W? W=X1+Xn X1 Xn iid, Bernoulli (p) PMF of W? MGF of W?
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
226 Midterm 2 April 18, 2013 8:0010:00pm Lucy Stone Hall Auditorium Chapters 34 Conditioning Random Variables Text Sections 2.9, 3.8, 4.9, 4.10 Conditioning by an Event 2.9 for discrete random variables 3.8 for continuous random variables 4.8 for
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
226 Midterm 2 April 18, 2013 8:0010:00pm Lucy Stone Hall Auditorium Chapters 34 Conditioning Random Variables Text Sections 2.9, 3.8, 4.9, 4.10 Conditioning by an Event 2.9 for discrete random variables 3.8 for continuous random variables 4.8 for
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Samples of Correlated X,Y XYsamples.m: XYsamples(rho,m) XYsamples(0,1000
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
226 Midterm 2 April 18, 2013 8:0010:00pm Lucy Stone Hall Auditorium Chapters 34 Conditioning Random Variables Text Sections 2.9, 3.8, 4.9, 4.10 Conditioning by an Event 2.9 for discrete random variables 3.8 for continuous random variables 4.8 for
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers SECOND EDITION Roy D. Yates David J. Goodman Definitions, Theorems, Proofs, Examples, Quizzes, Problems, Solutions Chapter 4
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Derived Random Variables X is a continuous random variable PDF CDF Find PDF and CDF of Y=g(X) PDF of Y Find the CDF = Take the derivative: = / Problem 1 is continuous uniform (0,1) = 10 + 20 Find the PDF of Problem 2 is continuous unifor
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers SECOND EDITION Roy D. Yates David J. Goodman Definitions, Theorems, Proofs, Examples, Quizzes, Problems, Solutions Chapter 4
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers SECOND EDITION Roy D. Yates David J. Goodman Definitions, Theorems, Proofs, Examples, Quizzes, Problems, Solutions Chapter 4
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Find the CDF FX(c) = P [ X c] X=65, (122,38) (122,38) P[X 65]? Derived Random Variables X is a continuous random variable PDF CDF Find PDF and CDF of Y=g(X) PDF of Y Find the CDF = Take the derivative: = / Problem 1 is continuous uniform (0
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers SECOND EDITION Roy D. Yates David J. Goodman Definitions, Theorems, Proofs, Examples, Quizzes, Problems, Solutions Chapter 3
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
CONTINUOUS RANDOM VARIABLE X P[X=x]=0 for all x CONTINUOUS RANDOM VARIABLE X P[X=x]=0 for all x CONTINUOUS RANDOM VARIABLE X P[X=x]=0 for all x Probabilities depend on slope of CDF: P[x1 < X x1 + ] < P[x2 < X x2 + ] Probabilities depend on slope of CDF: P
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Exponential rv X is memoryless: Wait time X (exponential ( =0.2) in minutes for the RU bus. Given you already waited x min, what is probability the bus comes now (in the next 1 sec) P[x < X x+1/60  X>x] = Uniform rv X is NOT memoryless: Wait time X (unif
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
226 Midterm 1 Feb 28, 2013 8:00pm Lucy Stone Hall Auditorium Livingston Three questions:
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Exponential rv X is memoryless: Wait time X (exponential (=0.2) in minutes for the RU bus. Given you already waited x min, what is probability the bus comes now (in the next 1 sec) P[x < X x+1/60  X>x] = Uniform rv X is NOT memoryless: Wait time X (unifo
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
H vIf{ e H H lllRUf lllRUf vSVe 0g jW 0 5d g 4`W a a C T{TXc3 ba a D D a 7 7 aD A D 9 C D D C DD D A g TB@7BTGwQ{3vTBGT{G(CG7BG4GQBc{GB{lGBDfa BTa `W Ifl4v e H lllRUf$ SV e SVe 2g jW 0 g 3jW
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
H First shot 1/3 2/3 G B Second Shot 3/4 3/4 1/4 1/4 G B G B (O) (O) (W) (L) PH 4 (3 cCvkT7cIzT(T@GfBccb4fT7BGI D AD7 CD7 D a D a a A D UH U PH RU 7 C 7 C a a 7 C CA AD b z@f@3 cCTGB@QzDGcVfXBTTGGCcCkTzcan8giW UH H
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
e DcCIjGT7 f AD gU gU g W @fHW @fHW f`2h Ca D TyT@cCT7 Y D b Aa 7 A ba Ca a C aD D A C 7 9DA C C 7 AD 9 b D 5G0TDfcafv@GBTatTjT8cCBTyBu(TjDT7yDcAByyDC@7dTmjuBTaT7GI AaD a aD D a D D b CD 7
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e 11 Denition 1.4 Axioms of Probability A probability measure P[] is a function that maps events in the sample space to real numbers such that Axiom 1 For any event A, P[A] 0. Axiom 2 P[S] = 1. Axiom 3 For any countable collection A1, A2,
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e Section 2.1 Tree Diagrams 35 Yates & Goodman 3e Problem 2.1.5 57 Suppose that for the general population, 1 in 5000 people carries the human immunodeciency virus (HIV). A test for the presence of HIV yields either a positive (+) or nega
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e 20 Theorem 1.8 For a partition B = cfw_B1, B2, . . . and any event A in the sample space, let Ci = A \ Bi. For i 6= j, the events Ci and Cj are mutually exclusive and A = C1 [ C2 [ . Yates & Goodman 3e Theorem 1.9 For any event A, and p
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
ECE#226# Course#Informa0on# Sakai# Everything#is#on#the#Sakai#site# Syllabus# Homework#Problem#Sets# Homework#Solu0ons# Student#Solu0on#Manual# Annotated#Lecture#Slides# Ask#Public#Ques0ons#in#the#sakai#chatroom# If#you#need#a#quick#reply,#send#me
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
PROBABILITY AND STOCHASTIC PROCESSES A FRIENDLY INTRODUCTION FOR ELECTRICAL AND COMPUTER ENGINEERS Third Edition Chapter 7 Viewgraphs Roy D. Yates & David J. Goodman Yates & Goodman 3e Section 7.1 Conditioning a Random Variable by an Event 242 Yates & Goo
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
PROBABILITY AND STOCHASTIC PROCESSES A FRIENDLY INTRODUCTION FOR ELECTRICAL AND COMPUTER ENGINEERS Third Edition Chapter 3 Viewgraphs Roy D. Yates & David J. Goodman Yates & Goodman 3e 62 Discrete Random Variables In this chapter and for most of the rema
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Yates & Goodman 3e 66 Probability Mass Function Denition 3.3 (PMF) The probability mass function (PMF) of the discrete random variable X is PX (x) = P [X = x] Yates & Goodman 3e 68 Families of Random Variables In practical applications, certain families
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
226 Midterm 1 Feb 28, 2012 8:00pm Lucy Stone Hall Auditorium Livingston
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Announcements Midterm 1: 2/28, 8:0010:00pm, Lucy Stone Hall Auditorium (LSH AUD) Covers material through the 2/20 lecture
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Monday, January 28, 2013 12:06 PM Lectture Page 1 Lectture Page 2 Lectture Page 3 Lectture Page 4 Lectture Page 5 Lectture Page 6 Lectture Page 7 Lectture Page 8 Lectture Page 9 Lectture Page 10 Lectture Page 11 Lectture Page 12 Lectture Page 13 Lectture
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Announcements Midterm 1: 2/28, 8:0010:00pm, Lucy Stone Hall Auditorium (LSH AUD) Covers material through the 2/20 lecture
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
kPermutation: ordered sequence of k objects Permutations vs Combinations Permutation: order of selection matters Choose a sequence of 3 cards out of 52 Combination: order of selection does not matter Choose 3 cards out of 52 n choose kkcombinati
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
A B C A B A B C D C CD D AB D A C D C D A B AB A A B C D C CD D C D C D Hi Ri i i Hi hidden I Ri reveals i Hi hidden I Ri reveals i
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Definition 2.4 Probability Mass Function (PMF) The probability mass function (PMF) of the discrete random variable X is PX(X)=P[X=X] Definition 2.5 Bernouiii (p) Random Variable X is a Bernoulli (p) random variable if the PMF of X has the form 1 p x = 0 P
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
14:332:226 Probability and Stochastic Processes Probability and Random Processes Random Process: Observe a random signal/waveform Examples: Traffic at a webserver Calls at a telephone switch Noise in a receiver Unemployed people Species population Y
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Oneoftheseisnotliketheothers. N=2400 random variables
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
X1,Xn iid: X1,Xn iid, Bernoulli (p) W= X1+Xn W=X+Y (X,Y independent) PDF of W? MGF of W? W=X1+Xn (X1 Xn independent) PDF of W? MGF of W? W=X1+Xn X1 Xn iid, Bernoulli (p) PMF of W? MGF of W?
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
226 Midterm 2 April 18, 2013 8:0010:00pm Lucy Stone Hall Auditorium Chapters 34 Conditioning Random Variables Text Sections 2.9, 3.8, 4.9, 4.10 Conditioning by an Event 2.9 for discrete random variables 3.8 for continuous random variables 4.8 for
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
226 Midterm 2 April 18, 2013 8:0010:00pm Lucy Stone Hall Auditorium Chapters 34 Conditioning Random Variables Text Sections 2.9, 3.8, 4.9, 4.10 Conditioning by an Event 2.9 for discrete random variables 3.8 for continuous random variables 4.8 for
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Samples of Correlated X,Y XYsamples.m: XYsamples(rho,m) XYsamples(0,1000
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
226 Midterm 2 April 18, 2013 8:0010:00pm Lucy Stone Hall Auditorium Chapters 34 Conditioning Random Variables Text Sections 2.9, 3.8, 4.9, 4.10 Conditioning by an Event 2.9 for discrete random variables 3.8 for continuous random variables 4.8 for
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers SECOND EDITION Roy D. Yates David J. Goodman Definitions, Theorems, Proofs, Examples, Quizzes, Problems, Solutions Chapter 4
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Derived Random Variables X is a continuous random variable PDF CDF Find PDF and CDF of Y=g(X) PDF of Y Find the CDF = Take the derivative: = / Problem 1 is continuous uniform (0,1) = 10 + 20 Find the PDF of Problem 2 is continuous unifor
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers SECOND EDITION Roy D. Yates David J. Goodman Definitions, Theorems, Proofs, Examples, Quizzes, Problems, Solutions Chapter 4
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers SECOND EDITION Roy D. Yates David J. Goodman Definitions, Theorems, Proofs, Examples, Quizzes, Problems, Solutions Chapter 4
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Find the CDF FX(c) = P [ X c] X=65, (122,38) (122,38) P[X 65]? Derived Random Variables X is a continuous random variable PDF CDF Find PDF and CDF of Y=g(X) PDF of Y Find the CDF = Take the derivative: = / Problem 1 is continuous uniform (0
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers SECOND EDITION Roy D. Yates David J. Goodman Definitions, Theorems, Proofs, Examples, Quizzes, Problems, Solutions Chapter 3
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
CONTINUOUS RANDOM VARIABLE X P[X=x]=0 for all x CONTINUOUS RANDOM VARIABLE X P[X=x]=0 for all x CONTINUOUS RANDOM VARIABLE X P[X=x]=0 for all x Probabilities depend on slope of CDF: P[x1 < X x1 + ] < P[x2 < X x2 + ] Probabilities depend on slope of CDF: P
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Exponential rv X is memoryless: Wait time X (exponential ( =0.2) in minutes for the RU bus. Given you already waited x min, what is probability the bus comes now (in the next 1 sec) P[x < X x+1/60  X>x] = Uniform rv X is NOT memoryless: Wait time X (unif
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
226 Midterm 1 Feb 28, 2013 8:00pm Lucy Stone Hall Auditorium Livingston Three questions:
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
Exponential rv X is memoryless: Wait time X (exponential (=0.2) in minutes for the RU bus. Given you already waited x min, what is probability the bus comes now (in the next 1 sec) P[x < X x+1/60  X>x] = Uniform rv X is NOT memoryless: Wait time X (unifo
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: NetID: 332:226 Signature: Probability and Stochastic Processes February 28, 2013 Examination 1 You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted.
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
332:226 Probability and Stochastic Processes April 18, 2013 SOLUTION (VERSION 1) (The solution for version 2 starts on page 7.) Examination 2 You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; nei
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: 332:226 NetID: Signature: Probability and Stochastic Processes February 26, 2015 Examination 1 You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted.
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
332:226 Probability and Stochastic Processes February 28, 2013 SOLUTION Examination 1A You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted. Make sure tha
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
 ,.,. A .i 332:226 Signature: _ _ _ _ _ _ _ Prob?bility and Stochastic Processes February 26, 2015 Examination 1 You have 110 minutes to answer the following questions in the:notebooks provided . This is. a c'losed book exam; neither notes nor calculator
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: NetID: 332:226 Signature: Probability and Stochastic Processes April 18, 2013 Examination 2 You have 110 minutes to answer the following questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted. Ma
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: 332:226 NetID: Signature: Probability and Stochastic Processes Final Examination May 9, 2013 MAKE SURE TO READ BOTH SIDES OF THIS EXAM PAPER. You have 180 minutes to answer the following questions in the notebooks provided. Make sure that you have i
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: NetID: 332:226 Probability and Stochastic Processes Calculus Prerequisite Quiz Quiz 0 You have 20 minutes to answer the following questions on this piece of paper. Calculators, textbooks, cheat sheets and other study aids are NOT permitted. Please
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
NAME: 332:226 NetID: Probability and Stochastic Processes Jan 2630, 2009 Quiz 1 You have 20 minutes to answer the following questions on this piece of paper. Calculators, textbooks, cheat sheets and other study aids are NOT permitted. Please make sure th
School: Rutgers
Course: PROBABILITY AND RANDOM PROCESSES
332:226 Probability and Stochastic Processes March 12, 2009 Examination 1 You have 110 minutes to answer the following four questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted. Make sure that you ha