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Unformatted text preview: EE 278 October 28, 2009 Statistical Signal Processing Handout #10 Homework #6 Due: Wednesday November 4 1. Gaussian random vector Suppose X ∼ N ( μ , Σ) is a Gaussian random vector with μ = 1 5 2 and Σ = 1 1 1 4 9 . a. Find the pdf of X 1 . b. Find the pdf of X 2 + X 3 . c. Find the pdf of 2 X 1 + X 2 + X 3 . d. Find the pdf of X 3 given ( X 1 ,X 2 ). e. Find the pdf of ( X 2 ,X 3 ) given X 1 . f. Find the pdf of 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 ....
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 Fall '09
 BalajiPrabhakar
 Variance, Signal Processing, Probability theory, GRV, Gaussian random vector

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