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Unformatted text preview: Jointly Continuous Random Variables III 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 34 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 2 of 41 Conditional pdf of X Introduction For a continuous random variable X , the conditional pdf f X  A (u  A) describes the probabilistic behavior of X given that the event A has occurred P{u X u+ u  A} = P[{u X u+ u} A]/P(A) f X  A (u  A) u In Lecture 29, we considered the case when A is specified in terms of X itself ECE 313  Lecture 34 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 3 of 41 X and Y are jointly continuous random variables, and suppose A = { Y } Given that A occurred, ( X , Y ) must be in the orange shaded region shown Conditional pdf of X given {Y } v u P{u X u+ u} = volume in green strip P[{u X u+ u} A] = volume in blue strip f X A (u  A) u vol. blue strip P(A) f X A (u  A) = f X , Y (u,v)dv F Y ( ) ECE 313  Lecture 34 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 4 of 41 The random point ( X , Y ) is uniformly distributed on region {(u,v): 0 < u < v < 1} f X , Y (u,v) = 2 for 0 < u < v < 1; green region Example D: conditional pdf given A u v 1 1 1/2 A = { Y 1/2}. P(A) = 1/4; orange region f X A (u  A) = f X , Y (u,v)dv P(A) 1/2 f X A (u  A) = 2(1/2u)/ (1/4) = 4(12u) for 0 < u < 1/2 u u ECE 313  Lecture 34 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 5 of 41 The random point ( X , Y ) is uniformly distributed on region {(u,v): 0 < u < v < 1} f X , Y (u,v) = 2 for 0 < u < v < 1; green region Example D: conditional pdf given A c u v 1 1 1/2 f X A c (u  A c ) = f X , Y (u,v)dv P(A c ) 1/2 = 4/3 for 0 < u < 1/2 = 8/3(1u) for 1/2 u < 1 = 0 otherwise A c = { Y > 1/2}. P(A c ) = 3/4; orange region ECE 313  Lecture 34 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 6 of 41 u v 1 1 1/2 f X , Y (u,v) = 2 for 0 < u < v < 1; green region f X (u) = f X A (u  A)P(A) + f X A c (u  A c )P(A c ) Example D: unconditional pdf 1/2 A c = { Y > 1/2}. P(A c ) = 3/4; orange region 1 u u u 4 f X A (u  A) 4/3 f X A c (u  A c ) 2 f X (u) ECE 313  Lecture 34 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 7 of 41 Conditional pdf of X: given {Y = } Suppose A = { Y = } instead of { Y } Now what is f X  A (u  A)?...
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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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