ECEN4503 Random SignalsECEN4503 Random SignalsLecture #4 22 January 2014Lecture #4 22 January 2014Dr. George ScheetsDr. George Scheets

ECEN4503 Random SignalsECEN4503 Random SignalsLecture #4 22 January 2014Lecture #4 22 January 2014Dr. George ScheetsDr. George Scheets

Read Chapter 2.7-3.1Read Chapter 2.7-3.1 Homework: Chapter 2: 20, 22, 25, & 26 (1Homework: Chapter 2: 20, 22, 25, & 26 (1stst Edition) Edition) Homework: Chapter 2: 51, 53, 58, 59 (2Homework: Chapter 2: 51, 53, 58, 59 (2ndnd Edition) Edition) Quiz #1 on 24 JanuaryQuiz #1 on 24 January

Quiz #1 Next TimeQuiz #1 Next Time Covers Chapter 2 (text)Covers Chapter 2 (text) < Random Variables (notes)< Random Variables (notes) Open Book, Notes, Instructor, & TAOpen Book, Notes, Instructor, & TA Closed NeighborClosed Neighbor For All Quizzes & Tests...For All Quizzes & Tests...

Calculators allowedCalculators allowed So long as I/O only keyboard & screenSo long as I/O only keyboard & screen

Smart Phones are Smart Phones are NOTNOT allowed allowed

ProbabilityProbability Bounded by [0,1]Bounded by [0,1] P(A + B) = P(A) + P(B) - P(AB)P(A + B) = P(A) + P(B) - P(AB) P(a|b) = P(ab)/P(b) P(a|b) = P(ab)/P(b) (Conditional Probability)(Conditional Probability) P(ab) = P(a)P(b) P(ab) = P(a)P(b) (Statistically Independent)(Statistically Independent) P(b) = P(aP(b) = P(a11b) + ... + P(ab) + ... + P(annb) (b) (Total Probability)Total Probability)

P( aP( a11 + ...+ a + ...+ an n ) must = 1) must = 1

aaii ∩ a∩ ajj must be mutually exclusive, i ≠ j must be mutually exclusive, i ≠ j

P(a|b)P(b) = P(b|a)P(a) P(a|b)P(b) = P(b|a)P(a) = P(ab) (= P(ab) (Baye's Theorem)Baye's Theorem)

Bernoulli Trials & Binomial DistributionBernoulli Trials & Binomial Distribution You must have a two state experimentYou must have a two state experiment

Success or FailureSuccess or Failure Stable probabilitiesStable probabilities

You must be interested in the You must be interested in the numbernumber of successes (or failures)of successes (or failures) Specific order not importantSpecific order not important

The individual experiments must be The individual experiments must be Statistically IndependentStatistically Independent

Need P(Number of Outcomes)?Need P(Number of Outcomes)? 1) Bernoulli Trial? Yes? Binomial Dist.1) Bernoulli Trial? Yes? Binomial Dist. 2) If no, think Conditional Probability2) If no, think Conditional Probability

Draw an event treeDraw an event tree Assign probabilitiesAssign probabilities

P(a1)

P(a2)

P(a3)

P(b1|a1)

P(b2|a1)

P(b4|a2)

P(b6|a3)

P(a1∩b1)

P(a2∩b3)

P(a1∩b2)

P(a2∩b4)

P(a3∩b5)

P(a3∩b6)

P(b3|a2)

P(b5|a3)