hwset4 - EE 278 Handout#12 Statistical Signal Processing...

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Unformatted text preview: EE 278 Handout #12 Statistical Signal Processing Monday, July 27, 2009 Homework Set #4 Due: Monday, August 3, 2004. 1. Given a Gaussian random vector X ∼ N ( μ , Σ), where μ = [ 1 5 2 ] T and Σ =    1 1 0 1 4 0 0 0 9    . Find the pdfs of the following random variables. (a) X 1 (b) X 2 + X 3 (c) 2 X 1 + X 2 + X 3 (d) X 3 given ( X 1 ,X 2 ) (e) ( X 2 ,X 3 ) given X 1 (f) X 1 given ( X 2 ,X 3 ). (g) Find P { 2 X 1 + X 2 + X 3 < } . (h) Find the joint pdf of Y = A X , where A = bracketleftBigg 2 1 1 1 − 1 1 bracketrightBigg . 2. Definition of Gaussian random vector In lecture notes #6 we defined Gaussian random vector via the joint pdf. There are other equivalent definitions, including the following very revealing definition. A random vector X with mean μ and covariance matrix Σ is a GRV if and only if Y = a T X is Gaussian for every real vector a negationslash = . In the lecture notes (Property 2) we showed that any linear transformation of a GRV results in a GRV. Thus the definition given in the lecture notes implies this new definition. In this problem you will prove the converse, i.e., that the new definition implies that the joint pdf of X has the form given in the lecture notes. You will do this using the characteristic function as follows: (a) Write down the definition of the characteristic function for X . (b) Define Y = ω T X . Note that the characteristic function of X reduces to the characteristic function of Y evaluated at ω = 1. (c) By the new definition, Y is Gaussian. Use this fact to write the characteristic function of X in terms of the mean and variance of Y . (d) Write down the mean and variance of Y in terms of ω and the mean and covariance matrix of X and substitute in the characteristic function of X . (e) Conclude that the joint pdf of X is Gaussian. 1 3. Proof of Property 4 In lecture notes #6 it was stated that conditionals of a Gaussian random vector are Gaussian. In this problem you will prove that fact: If bracketleftBigg Y X bracketrightBigg is a 0-mean GRV then X |{ Y = y } ∼ N (Σ X Y Σ- 1 Y y , σ 2 X − Σ X Y Σ- 1 Y Σ Y X ) ....
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This note was uploaded on 10/26/2011 for the course ECE 180 taught by Professor Eggers during the Spring '11 term at Aarhus Universitet.

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hwset4 - EE 278 Handout#12 Statistical Signal Processing...

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