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Unformatted text preview: Jointly Gaussian Random Variables Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign. All Rights Reserved ECE 313 Probability with Engineering Applications ECE 313  Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, 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 ECE 313  Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, 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 ECE 313  Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, 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 ECE 313  Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, 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 ECE 313  Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 6 of 41 2 pictures = 2000 words: Part I X , Y independent zeromean 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 ECE 313  Lecture 39 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, 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...
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This note was uploaded on 09/29/2009 for the course ECE 123 taught by Professor Mr.pil during the Spring '09 term at University of Iowa.
 Spring '09
 mr.pil

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