Lecture17_ExpectedVectorJointGaussian

Lecture17_ExpectedVectorJointGaussian - ECE 340...

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ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Prof. Vince Calhoun Prof. Vince Calhoun Lecture 17: Vector RV Lecture 17: Vector RV ’s, Jointly s, Jointly Gaussian RV Gaussian RV ’s
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Quiz #7 •G i v e n Z = X + Y , f x (x) and f y (y) write an expression for the pdf of Z if X & Y are independent. i v e n F ( x 1 ,x 2 ,….,x N ), write an expression for the marginal cdf, F(x 1 ,x 2 )
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• Section 6.1-6.2 • Functions of several RV’s • Section 6.3-6.4 • Expected value of vector RV’s • Jointly Gaussian RV’s
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pdf of Linear Transformations We consider first the linear transformation of two random variables : or Denote the above matrix by A . We will assume A has an inverse, so each point ( v , w ) has a unique corresponding point ( x , y ) obtained from In Fig. 4.15, the infinitesimal rectangle and the parallelogram are equivalent events, so their probabilities must be equal. Thus eY cX W bY aX V + = + = . = Y X e c b a W V (4.56) . 1 = w v A y x dP w v f dxdy y x f W V Y X ) , ( ) , ( , ,
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where dP is the area of the parallelogram. The joint pdf of V and W is thus given by where x an y are related to ( v , w ) by Eq. (4.56) It can be shown that so the “stretch factor” is where | A | is the determinant of A . Let the n-dimensional vector Z be where A is an invertible matrix. The joint of Z is then (4.57) , ) , ( ) , ( , , dxdy dP y x f w v f Y X W V = ( ) , dxdy bc ae dP = ( ) () , A bc ae dxdy dxdy bc ae dxdy dP = = = , X Z A = n n ×
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EXAMPLE 4.36 Linear Transformation of Jointly Gaussian Random Variables Let X and Y be the jointly Gaussian random variables. Let V and W be obtained from ( X , Y ) by Find the joint pdf of V and W. |A| = 2, () ( ) ( ) (4.58) z z x Z A A f A x x f z z f f z A x n X X n Z Z n n 1 1 , , 1 , , 1 1 1 , , , , ) ( = = = .
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Lecture17_ExpectedVectorJointGaussian - ECE 340...

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