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by: Cindy Nguyen

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# ELEG 310 Week 8 Notes ELEG310

Cindy Nguyen
UD

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Week of 4/5 - 4/7
COURSE
Random Signals and Noise
PROF.
Dr. Daniel Weile
TYPE
Class Notes
PAGES
5
WORDS
CONCEPTS
eleg, eleg310, random signals and noise, Probability
KARMA
25 ?

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This 5 page Class Notes was uploaded by Cindy Nguyen on Saturday April 9, 2016. The Class Notes belongs to ELEG310 at University of Delaware taught by Dr. Daniel Weile in Spring 2016. Since its upload, it has received 29 views. For similar materials see Random Signals and Noise in Electrical Engineering at University of Delaware.

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Date Created: 04/09/16
ELEG 310 Week 8 Notes Topic 1: Multiple Random Variables Recall marginal pdf’s and cdf’s. Example: a jointly exponential pdf (density) ????????????(????,???? = ???????? −????????????−???????? 0 ≤ x,y ≤ ∞ Find: a) C b) marginal pdfs c) P[X<Y] d) P[X + Y < k] a) Integrate, make sure it is 1 ∞ ∞ = ∫ ∫ ???????? −????????????−???????? ???????? ???????? 0 0 −???????? −???????? ???? ∞ ???? ∞ = ???? [ ???? ]0 [ −???? ] 0 ???? = ???????? = 1 ????ℎ????????????????????????????,???? = ???????? ∞ −???????? −???????? b) ???????????? = ∫0 ???????????? ???? ???????? −???????? = ???????????? −????????[???? ]∞ ???? 0 = ???????? −???????? *integral must be positive c) P[X < Y] ∞ ∞ = ∫ ∫ ???????????? −???????? −???????????????? ???????? ????=0 ????=???? ∞ −???????? = ∫ ???????????? −???????? [ ]∞ ???????? −???? ???? ????=0 ∞ ????−???????? ∞ = ???? ∫ ???????? −???????? [ ] ???????? −???? ???? ????=0 ∞ = ???? ∫ ???? −(????+????)???????????? ????=0 −(????+????)???? ???? ∞ = ???? [ −(???? + ????) 0 ???? = ???? + ???? d) P[X + Y < k] ???? ????−???? −???????? −???????? = ????=0 ∫????=0???????????? ???? ???????? ???????? = ???? ????−???????? ???? − ???? = ∫ ???????????? −???????? [ ] ???????? −???? 0 ????=0 ???? = ???? ∫ ???? −????????[1 − ????−????(????−????] ???????? ????=0 ???? = ???? ∫[???? −????????− ????−????????????(????−????)????] ???????? ????=0 −???????? (????−????)???? ???? −???????????? ???? = ???? [ −???? − ???? ???? − ????] ???? −???????? −???????? −???????? ???? − ???? = 1 + ???? + ???? − ???? Independence ???? ???? ∈ ???? ,???? ∈ ???? ] = ???? ???? ∈ ???? ] ????[ ???? ∈ ???? ] 1 2 1 2 Suppose X and Y are independent discrete random variables. Let event???? = {???? ∈ ???? } 1 ???? ????2= ???? ∈ ???? ????} Pmf  ???? (???? ,???? ) = ????[???? ∈ ???? ,???? ∈ ???? ] by definition ????,???? ???? ???? ???? ???? = ????[???? ∈ ???? ]????????[???? ∈ ???? ???? by definition independence ( ) = ???? ???????? ???????? ???? ???? by definition of one-dimensional pmf Type equation here. Proof Let A = ????1∩ ???? 2 P[A] = ∑ ???? ∈ ????∑ ???? ∈ ???? ????????,????(????????,????????) ???? 1 ???? 2 = ∑????????∈ ????1∑????????∈ ????2 ????????(????????)????????????( ????) = ∑????????∈ ????1????????(????????) ∑ ????????∈ 2 ???????????? ????) Joint pmf and marginal pmf = ????[???? ]????[???? ] 1 2 Discrete Example Recall problem with message in bytes, broken into blocks of length M (see previous note set) P[Q = q, R = r] = P[N = qM + r] qM + r = (1 – p)p M Mq P[Q = q] = (1 – p )p 1−???? ???? P[R = r] = ???? ???? 1−???? 1−???? P[Q = q] P[R = r] = (1 – p )p Mq ???????? 1−???? ???? = (1 – p)pqM + r Example: Independent joint distribution is the product of the marginal distribution. Let event????1= ???? ≤ ???? } ????2= ???? ≤ ???? } ???? ????,???? = ???? ???? ∩ ???? ] ????,???? 1 2 = ????[????1]????[???? 2 = ???????????? ???? ????( ) also, ????,????????,???? = ???? ???? ???? ???????? ) ???? ???? ∈ ???? ,???? ∈ ???? ]= ∫ ∫ ???? (????,???? ???????? ???????? 1 2 ????,???? ????∈ 1 ????∈2 = ∫ ???? ???? ???????? ∫ ???? ???? ????????( ) ????∈ 1 ????∈????2 = ???? ???? ∈ ???? 1 ????[???? ∈ ???? 2] New Example Z = g(X,Y) Continuous: ∞ ∞ ???? ???? = ∫ ∫ ????(????,????)???? (????,???? ???????? ???????? ????,???? −∞ −∞ “Sum over all possible values.” Discrete: ∞ ∞ ???? ???? = ∑ ∑ ????(???? ???????? ???????? ????,????????????,????????) −∞ −∞ Let Z = X + Y. Find E[Z] ∞ ∞ ′ ′ ???? ???? = ∫ ∫ (???? + ???? )????,????????,???? ???????? ???????? −∞ −∞ (internal integral first) ∞ ∞ ∞ ∞ = ∫ ???? [∫ ????????,????????,???? ???????? ????????]???????? + ∫ ????[∫ ????????,????(????,???? ???????? ????] ???????? −∞ −∞ −∞ −∞ ∞ ∞ = ∫ ???? ???????????? ???????? + ∫ ???? ???? ???? ???????? −∞ −∞ = E[X]E[Y] th The jk central moment (centered on the mean) j k E[(X – E[X]) (Y – E[Y]) Covariance: COV[X,Y] = E[(X – E[X]) (Y – E[Y])] We can use the simplified version: COV[X,Y] = E[ XY – XE[Y] – YE[X] ] + E[X]E[Y] = E[XY] – E[X]E[Y] – E[Y]E[X] + E[X]E[Y] = E[XY] – E[X]E[Y] Suppose X and Y are independent. The joint pdf is the product of the marginal pdf. Correlation Coefficient: COV[X,Y] ????????????[????,????] PX,Y = ???????? ????

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