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ECE313.Lecture39 - ECE 313 Probability with Engineering...

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Jointly Gaussian Random Variables Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications
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ECE 313 - Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 41 Independent unit Gaussian RVs Let W and Z denote independent unit Gaussian random variables f W ( α ) = (1/2 π ) 1/2 •exp(– α 2 /2) = φ ( α ) f Z ( β ) = (1/2 π ) 1/2 •exp(– β 2 /2) = φ ( β ) where φ (•) is the unit Gaussian pdf f W , Z ( α , β ) = f W ( α )•f Z ( β ) = φ ( α )• φ ( β ) = (1/2 π )•exp[–( α 2 + β 2 )/2] This is a circularly symmetric pdf
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ECE 313 - Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 41 Independent Gaussian RVs W and Z independent N (0, 1) RVs Let X = σ X W + μ X Y = σ Y Z + μ Y X and Y are N ( μ X , ( σ X ) 2 ) and N ( μ Y , ( σ Y ) 2 ) random variables respectively because linear functions of Gaussian RVs are Gaussian RVs X and Y are independent RVs because they are functions of independent RVs
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ECE 313 - Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 41 pdf of independent Gaussian RVs f X (u) = (1/ σ X )• φ ((u– μ X )/ σ X ) f Y (v) = (1/ σ Y )• φ ((v– μ Y )/ σ Y ) f X , Y (u,v) = f X (u)•f Y (v) = (1/ σ X )• φ ((u– μ X )/ σ X )•(1/ σ Y )• φ ((v– μ Y )/ σ Y ) = C•exp { ( [(u– μ X )/ σ X ] 2 + [(v– μ Y )/ σ Y ] 2 ) /2 } where C = 1/(2 πσ X σ Y ) This is a not a circularly symmetric pdf The curve f X , Y (u,v) = c is an ellipse
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ECE 313 - Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 41 The contours are ellipses X and Y are independent N ( μ X , ( σ X ) 2 ) and N ( μ Y , ( σ Y ) 2 ) RVs f X , Y (u,v) = f X (u)•f Y (v) = C•exp { ( [(u– μ X )/ σ X ] 2 + [(v– μ Y )/ σ Y ] 2 ) /2 } where C = (1/2 πσ X σ Y ) The curve f X , Y (u,v) = c defines an ellipse that is centered at ( μ X , μ Y ) and whose axes are parallel to the coordinate axes
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ECE 313 - Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 41 2 pictures = 2000 words: Part I X , Y independent zero-mean Gaussians with variances ( σ X ) 2 and ( σ Y ) 2 ( σ X ) 2 > ( σ Y ) 2 u v v ( σ X ) 2 < ( σ Y ) 2 u
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ECE 313 - Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 41 Jointly Gaussian random variables The random variables in the previous few slides are simple examples of what are called jointly Gaussian random variables The RVs are independent and their marginal pdfs are Gaussian pdfs Generally, if W and Z are independent N (0, 1) RVs, then X = a W + b Z + μ X and Y = c W + d Z + μ Y are called jointly Gaussian random variables
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ECE 313 - Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 8 of 41 Linear transformations If W and Z are independent N (0, 1) RVs, then X = a W + b Z + μ X , Y = c W + d Z + μ Y are called jointly Gaussian RVs
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