ECE313.Lecture33

ECE313.Lecture33 - Jointly Continuous Random Variables II...

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Unformatted text preview: Jointly Continuous Random Variables II 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 ECE 313 - Lecture 33 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 35 Jointly continuous RVs Jointly continuous random variables spread the probability mass (usually with varying density ) over a region The joint pdf f X , Y (u,v) tells us how dense the probability mass is at the point (u,v) There is no probability mass at any point P{( X , Y ) = (u,v)} = P{ X = u, Y = v} = 0 for all real numbers u and v f X , Y (u,v) is the density of the probability mass Units are probability mass per unit area ECE 313 - Lecture 33 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 35 The joint pdf is mass per unit area The joint pdf f X , Y (u,v) is not a probability: we must multiply the pdf by an area to get a probability P{( X , Y ) { small region containing (u,v)}} f X , Y (u,v){area of region} For larger regions, Probability = (double) integral of the pdf over the region P{( X , Y ) A} = f X , Y (u,v) du dv A ECE 313 - Lecture 33 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 35 Graphical interpretation P{( X , Y ) { small region containing (u,v)}} f X , Y (u,v){area of region} volume above the u-v plane and below the f X , Y surface u v Joint pdf is a surface above the u-v plane height f X , Y (u,v) base area height base area ECE 313 - Lecture 33 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 35 Probability = volume under joint pdf P{( X , Y ) A} = f X , Y (u,v) du dv A P{u X u+ u,v Y v+ v} f X , Y (u,v) u v is the volume of a prism of height f X , Y (u,v) and rectangular base of area u v P{( X , Y ) A} = volume of solid with vertical sides, base A, and varying height ECE 313 - Lecture 33 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 35 Properties of the joint pdf f X , Y (u,v) 0 for all u and v Total volume between the f X , Y surface and the u-v plane is 1 This is just the interpretation of the result F X , Y ( , ) = 1 = f X , Y (u,v) du dv v= u= F X , Y (u ,v ) = f X , Y (u,v) du dv or integrate w.r.t. v rst and then w.r.t. to u v= v u= u ECE 313 - Lecture 33 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 35 Finding joint pdf from the joint CDF f X , Y (u,v) = F X,Y (u,v) if the derivative exists, and f X , Y (u,v) = any number 0 otherwise...
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ECE313.Lecture33 - Jointly Continuous Random Variables II...

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