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# ELEG 310 Week 9 Notes ELEG310

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This 5 page Class Notes was uploaded by Cindy Nguyen on Sunday April 17, 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 27 views. For similar materials see Random Signals and Noise in Electrical Engineering at University of Delaware.

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Date Created: 04/17/16

ELEG 310 Week 9 Notes Definition of joint moment (where X and Y are jointly continuous): ????[???? ???? ] = ∫∞ ∫∞ ???? ???? ???? (????,???? ???????? ???????? −∞ −∞ ???????? Covariance: COV[X,Y] = E[XY] – E[X]E[Y] Correlation Coefficient: PX,Y= ????????????[????,????] where ????????= √????????????(????), ???? ???? √ ????????????(????) ???????? ???? Example: Y = aX+ b E[Y] = aE[X] + b VAR[Y] = a VAR[X] E[X,Y] = E[X(aX + b)] =aE[X ]+ bE[X] ???? = ????????????[????,????]= E[XY] – E[X]E[Y] correlation coefficient X,Y ???????? ???? √????????????(????√ ????????????(????) a[E[???? ] – E[????] ] ???? = = = sign(a) |????|????????????[????] |????| Independence If COV(X,Y) = 0, the pair of random variables are independent, and uncorrelated. X and Y are uncorrelated if ???? X,Y = 0. Therefore, if COV(X,Y) = 0, then ???? = 0, because independence implies X,Y they are uncorrelated, but NOT the other way around. Example of being uncorrelated, but also not independent: ???? ∈ [0,2????] (theta is uniformly distributed in 0 to 2????) ???? = cosθ ???? = sinθ Is the joint distribution the product of the marginal distribution? ????θ ???? ???? = ∑???? ????(θ) |θ =???????? ???????? ”inverse image”; all x whose cos are the given ????; All solutions where cos ???? = ???? Easier to compute????????= -sinθ = −sinθ (????????????1(???? ) ????θ ???????? = -sinθ = −sinθ (????????????(???? ) ????θ ???????? 2 ????θ = −√1 − ???? for???? 1 2 2 sin???? 1= √1 − ???? sin????2= − √ 1 − ???? Plug using formula above 1 1 1 ???????????? = ( 2 + 2) 2???? √1 − ???? √1 − ???? = 1 marginal pdf of x ???? 1−???? 2 ???? ???? ) = 1 same marginal distribution ???? √ 2 ???? 1−???? ????????(????)????????(????) = ????????????(????,???? ) (Marginal distribution) (Joint distribution) Does product of marginal distribution = joint distribution? No. Graphical representation: (marginal rep. by red circle; joint rep. by blue square) Clearly, not independent. Are they correlated? Recall: independence implies uncorrelated, but NOT the other way around. 2???? 1 E[X] = E [sinθ] =∫0 sinθ(2???????????? = 0 Appropriate density o E[Y] = 0 (same as above, just shifted 90 ) E[XY] = ∫ 2????sinθ????????????θ( )???????? 0 2???? Sin(x)cos(x) means it’s orthogonal There fore = 0 Conditional pdf Continuous distribution ????[???? ∩ ????=???? ] P[Y ∈ ???? ???? = ???? = ] ????[????=????] ???????????? ???? = lim ???? ???? ???? <???????? ≤ ???? + ℎ) ℎ→0 This means that big X fills in small interval around little x ????[{????=????} ∩ ????< ????≤ ????+ℎ ] By definition = lim ℎ→0 ????[????< ????≤ ????+ℎ] ???? ????+ℎ ∫−∞ ???? ????????????(????,???? ???????? ???????? = lim ????+ℎ ℎ→0 ???? ???????????? ???????? ℎ∫???? ???? (????,????′ ????????′ = lim −∞ ???????? (x is a constant, y’ is y prime) ℎ→0 ℎ????????(????) ∫???? ????????????(????,????′ ????????′ = −∞ (definition) ????????(????) Therefore, the definition of conditional pdf: ???? ????????????????,????) ???????????? ????) = ???? ???? ???? = ???????? ????????(????) For distribution in Y, X is a parameter: P[Y = something given we know (we know x = something)] ???? (????,????) = ???? (????|????) ???? (????) ???????? ???? ???? Example Let there be uniform distribution. Pick X ∈ [0,1] and Y ∈ 0,1 .[ ] So Y depends on X. Find cdf and pdf of y Solution: Suppose X=x. ????/???? ????>???? P[Y≤y] | X=x] = {1 ????<???? Cdf: ???? ???? = ????[???? < ????] ???? ???? = ∫0 ???? ???? < ???? ???? = ????]????[???? = ????] (like always, except it’s an integral instead of sum) = ∫????1 ???????? + ∫ 1 1???????? 0 ???? ???? = ???? ]????+ ???? ???????????????? ]1 0 ???? = ???? − ???????????????????? ???? So ????(????) = ???????????? ) ???????? 1 = 1 − ???????????????? + ???? 1 = 1 − ???????????????? + ???? = −???????????? ????

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