This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 UrbanaChampaign. All Rights Reserved ECE 313 Probability with Engineering Applications ECE 313  Lecture 33 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, 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 UrbanaChampaign, 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 UrbanaChampaign, 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 uv plane and below the f X , Y surface u v Joint pdf is a surface above the uv plane height f X , Y (u,v) base area height base area ECE 313  Lecture 33 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, 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 UrbanaChampaign, 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 uv 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 UrbanaChampaign, 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...
View Full
Document
 Spring '09
 mr.pil

Click to edit the document details